Dynamic Differential Systems [I, 1 ed.] 9789730395525

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DYNAMIC DIFFERENTIAL SYSTEMS PART ONE SHPELESS MATTER DYNAMICS PART TWO DYNAMICS OF BIOLOGICAL AND ECONOMIC PROCESSES

VOLUME I Translator Ciupa Anca

Timișoara 2023







ISBN 978-973-0-39552-5





Prof. Dr. VIRGIL OBĂDEANU



DYNAMIC DIFFERENTIAL SYSTEMS PART ONE SHPELESS MATTER DYNAMICS PART TWO DYNAMICS OF BIOLOGICAL AND ECONOMIC PROCESSES

VOLUME I Translator Ciupa Anca

Timișoara 2023 ISBN 978-973-0-39552-5







2





Prof. Dr. VIRGIL OBĂDEANU

Translator Ciupa Anca Copyright © 2023 Flavius Ungureanu

E-mail: ungureanu [email protected]

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Il libro della natura e scritto nella lingua di matematica Galileo Galilei FOREWORD According to the knowledge we have at the moment, as well as to our capacity of understanding, the universe we live in can be thought as being made up and organized in three stages (levels) of development: shapeless matter, life and consciousness. The irst of these levels, the shapeless matter, is the simplest and to a good extent, accessible to study. The second, obviously more complex, is given, in some sort of expression, as to understand it. For us, people, being on the third stage, this level is far from being understood and explained, with the knowledge and language we have, at least at this level of development. Consequently, being endowed with consciousness, we ask ourselves questions and seek answers (to some of them), especially to the troublesome problem of our own existence. One question is: which side of the universe, called system, is changing, has an evolution? An answer to this kind of questions tried to ind, chronologically speaking, in turn: the ancient civilisation (through myths), religion (by faith), philosophy (by reasoning) ad science (by experiencing). In this irst part of the book, we shall limit, almost exclusively, only to the evolution of shapeless matter. In developing the study of dynamic systems, there should be considered certain principles, of philosophic order, on which theory is based, as well as certain formalisms, of mathematical order, adequate to the problems imposed to be solved. All these represent certain facades and modalities of expression and study of the real systems, via various models. The causalit-determinist principle expresses our conviction that the current condition of a given system is the effect of a previous condition called its cause. In its turn, this condition, is the cause of a future condition of the system, as effect. The inalist principle says that the evolution of the system taken up for study is done on a certain purpose and that is, such as to consume the smallest quantity of „action” which is constituted as the product of the multiplication between energy and time. Another point of view is the conservative, or invariant one. It says that the evolution of the system is done such as certain „somethings” would not change, would remain invariant. The book has a deep philosophical character. It presents the impact between faith (religion, philosophy and science, an impact presented by the principles: causalist-determinist, inalist and conservative, by their identity. The name “dynamic system” is ambiguous and many times confusing, due to the ample developments and generalizations, as well as to their applications. In the classical sense, the Greek sense, the notion “dynamics” is translated as force, power. Yet, throughout history, it suffered certain transformations by generalization, and today we perceive it as representing a change, an evolution. In the current expression, it entered almost all ields of thinking and activity. Basically, a “real dynamic system” is a part of the Universe, characterized by a inite number of parameters: (properties, features) expressed by real or complex, called condition parameters, whose values change in time and of which we say they constitute the evolution in time of the respective system. We shall understand, until identi ication to it, its various mathematical models. These values are seen differently by different observers found themselves in different reference systems (observers, that to smaller or larger extend, in luence themselves, the evolution of the system considered, via command parameters). Since the evolution of any system is in luenced by its interference with its complementary in the Universe, it essentially depends on the complementary. We shall understand, until the identi ication to it, its various mathematical models. Anyway, these models are expressed directly or indirectly, by differential equations or by partial derivatives and they are submitted to certain formalisms. The faith man feels, in their conviction that in this world (almost) “everything” is in motion, is changing, suggests that evolution is done such as certain “somethings” would not change, would remain invariant to time pass, and would preserve. This we shall call the invariant or conservative

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principle. Last, what doesn’t change, must be probably, the “laws according to which change occurs”, the evolution laws and especially the conservation laws of those somethings (that don’t change). A part of these last ones, in mathematics, are known and expressed as prime integrals. Philosophy has the point of view according to which the evolution of any system is done such as to achieve a inal purpose. T claims that the evolution of systems in general and of the mechanical ones in particular, is done on a inal purpose, that is such as with passing of the condition from any given moment to any other subsequent moment, to spend minimum action. This principle is known as the principle of minimum action or Hamilton principle, according to which evolution is open by conveying a function, called Lagrange function. The formalism we meet within these ideas is called the Lagrange formalism. Science, through its experimental character, and theoretical at the same time, claims and says that the evolution of dynamic systems is expressed through causal relations, as the effect of certain causes that it tries to ind and underline. A system evolves due to the in luence of its complementary in the universe (as well as to its various parts one over the other). This evolution is characterized by the change of some of its characteristics in time. Such changes are perceived by observing and measuring them as well as their speeds of changing etc. As known, when studying dynamic systems, various formalisms are used, such as: the Newtonian, Lagrange, Hamiltonian, Brinkhoff, etc. Each of these is adequate to certain problems it tries to solve. A real evolution process is modelled, thanks to mathematics, via an equation system, usually differential or with partial derivatives. The philosophical nature of this modelling represents the causalist – determinist principle, on which science is based. Consequently, knowing the current condition of a system is the only measure to foresee the condition of the system at any other subsequent moment. In mechanics, the body position changes are appreciated by the speeds and accelerations that these bodies receive. The expression conveyed to this concept is done by its modelling via a second degreedifferential equations, written in a form called “kinematic”, by Isaac Newton. The formalism that science (mechanics) uses is known as Newtonian formalism od mechanics. The ideas and dynamic notions born and initially developed within the study of physics, entered via generalization into the study of biology, as well as of that of economy, constituting thus new research ields: “biodynamics”, respectively “economic mechanics”. Science (classical) expresses the idea that within the evolution of a mechanical system, its current condition is the effect of a cause, one of its previous conditions (being of a causalist nature). Thus, we obtain the determinist expression, equal to the causalist one, which says that knowing the condition of a system in a certain moment in time determines n a certain way, the existence and knowing the condition in any other previous or subsequent moment to that from where we started. In a way, the current condition of a system is the effect that has as cause its previous condition, and its future condition is the effect of its current condition (respectively f any other previous condition). Their modelling is done, within the evolution of shapeless matter, via mathematical formalisms: Newtonian, LaGrange, Hamiltonian, Brinkhoff and Noetherian (invariant). Each of them is adequate to certain problems that they try to solve (in this volume). There comes the natural question: which of these three points of view is the most general, the most comprehensive. It is known that if given the evolution equations as Newton equations [8], they imply the existence of a set of prime integrals (laws of conservation), so that the conservative principle is a consequence of the cuasalist one, but reciprocally, knowing a set of independent prime integrals determines evolution, from where it results that the two points of view correspond perfectly to one another. N the other hand, de ining evolution through the principle of minimum action, following the existence of a Lagrange function, it results that this evolution is done according to the Euler-Lagrange equations, equal to a Newtonian equation system. This problem is known as the direct problem of mechanics, and thus any system evolving according to the inalist principle may be seen as evolving by observing the causalist principle, so any inalist system is also causalist. Raising and solving the reciprocal to the problem above expressed, and namely showing to what extent a system evolving from in a causalist manner can be seen as evolving in a inalist manner, was a very dif icult problem. It bears the name of the inverse problem of mechanics. Finally, even this problem was af irmatively, positively solved. Consequently, the three points of view are fully equal. To conclude with, (for the moment) the three vast, equal ields represent three facades of how we can see and conceive nature (the world) such as it is, unique. This unitary view of nature, with roots much older, was presented by the philosopher Jean Guiton and called, at the end of the 20th century, materialism. [J, G]

i = f i (t, ). or with partial derivatives, such as in the case of deformation continuous media evolution, or of the evolution of ields. In the study of certain dynamic systems in general and of those above-mentioned in particular, a few major problems are targeted, among which we quote: First of all inding laws (chapter VII) governing the evolution of systems, laws that are usually expressed locally, in the language of differential equations, or globally in that of vectoral ields, their integration, inding of general solutions and respectively of certain particular solutions corresponding to certain initial conditions or given to the limit, the quality and quantity study of these solutions, stability problems. In establishing certain invariant properties, among which we centrally ind the prime integrals or (more generally), the conservation laws. We retain here that one of the purposes we target in this irst part, together with that of inding the governing differential equations, or that of inding Lagrange or Hamilton functions, is also that of obtaining conservation laws, evolution prime integrals, that are expressed by real value functions, de ined on the condition space (or even more generally on the evolution space), dependent also on time, constant during the evolution of the system. To determine such laws, we often turn to the inalist principle. According to this, evolution happens in such a way that certain characteristic sizes of the system, and not few in numbers, be minimum during the evolution from one condition to another. Today we know the equality of the two main modalities of describing evolution. These latter ones supply though one of the mathematical modalities of inding conservation laws: Norther Theorem (chap. VII). All these come under the points of view: causalist-determinist, or inalist, of the dynamic system theory, points of view that shall constitute the content of this volume. Secondly, when deducting the evolution equations from the inalist principle (Chap. II), that consists in inding a Lagrange function corresponding to the system and writing the evolution equations as the Euler-Lagrange expression, material contained by Chap. II, and the association to LaGrange formalism and to the Hamiltonian formalism (Chap. III), respectively determining the Hamilton function, the Hamilton equations and expressing the Hamilton-Jacobi equations (Chap. IV). A fundamental purpose of studying dynamics in general and biodynamics in particular, is represented by determining the inite evolution laws, from the differential laws and, implicitly, of certain singularities. Such an approach is not always possible and easily done. The sixth chapter generalizes the above-expressed formalisms, by presenting a new formalism, that of Brinkhoff’s. The major purpose of this chapter VI is to show that for any dynamic system whose evolution is described by the second degree differential equations (Newtonian), its evolution may be described by a Lagrange function, dependent on acceleration, linear in its components, so that the Euler-Lagrange equations be Newtonian equations and their derivatives (§ 6.4).

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When studying certain concrete problems, the above-mentioned formalisms interweave often, due to what we can obtain new results, surprising, that could have not been found, so easily at least, by any other formalism deemed separately. The mathematical expression of the causalist principle has as starting point Isaac Newton’s work of 1686. The birth of LaGrange formalism, corresponding to the inalist principle, is marked by the expression of minimum time in optics, by Pierre de Fermat, in 1657 and it is deemed closed by solving the inverse problem by Brinkhoff [4] in 1927. The book aims to show, in the irst part, the detailed connections among these three “facades” (of the same reality) under which we are presented the evolution studies of the same unique nature, called system and constituted a part of the Universe. It is dedicated to the study of differential dynamic systems, organized per principles: causalist – determinist, inalist and respectively, conservative. It is known that dynamic systems represent an important chapter in the study of systems, especially in that of dynamic systems with applications, as we already speci ied, in mechanics, physics, biology and economy, being paced on the border between mathematics and other types of sciences, giving birth to new branches as bio-mathematics or economic mechanics, etc. thus, some of the mathematical models encountered in the study of mechanics can be successfully used, not only in physics but in biology, economy and other ields, as we shall see in the second part. Such study methods contain dynamic systems whose evolution can be described, for instance, by differential equations, presented in local form by:

As it is well-known, a Lagrange function was deemed de inite on , and the corresponding Euler-Lagrange equations are second degree equations. Not any second-degree equation system is expressed as a Euler-Lagrange equation system. This reciprocal does not occur for second-degree systems in general. Yet, if a Lagrange function, linear in speed components, then the corresponding Euler-Lagrange equations constitute a irst-degree system, linear in speed components and reciprocally. The demonstration of the reciprocal problem was initially done via generalization, by passing from the Lagrange function to Hamilton function. Hamilton equations were then generalized and returning to the tangent variety, a corresponding Lagrange function was found. The book presents, by singularization, the same property. Any second-degree equation system is actually a special irst-degree system on the tangent variety. Being a irst-degree system on TM, it has as corresponding a Lagrange function linear (af ine) in the speed component. In conclusion, the formalisms necessary to present this unit theory, and presented above, and essentially used in the text, are the Newtonian, LaGrange, Hamiltonian, Hamilton – Jacobi, Routhian, Brinkhoff and Noetherian. Aiming to apply the theoretical results, in the study of certain concrete systems, exact calculation methods do not work. To obtain information, an approximate original calculation method is presented. The book addresses those who study natural sciences, using mathematical methods. By the examples contained and solved to a good extent, we hope that the material presented be useful to the specialists in these ields. A “real” dynamic system is a part of the universe, characterized by a number of properties expressed by just as many real characteristic values, variable in time (for instance, the position in space, weight, colour, speed, acceleration, etc) and expressed by real numbers. All these being variable in time, and in turn, inter-connected. Putting these values in correspondence can be done by expressing them in numbers, real or complex, from an arithmetical space (differentiable variety) and expressed by a function F, de ined on a value system with values in differentiable variety whose image is expressed on a local map. The core of the injective application, to a real system to the arithmetical space is noted as values bearing the name of “model” of the real dynamic system with which the real dynamic system is identi ied. Such a model is expressed mathematically, thus via a differential equation system, or more in general, by partial derivatives with which the real system is identi ied. The mathematical model of a real dynamic system is not unique. The abstract notion of system, in its current understanding, assumes a multitude of objects (elements, or more general parts of an assemblage) in luencing one another and upon which external forces are exerted. In their turn, these elements exert outwards action. Among the fundamental properties that any system has, we retain that of transformation, as a possibility of being in luenced, via convenient choices of the actions exerted on the system from outside. To study nature, and more precisely a part of it, which for us represents a system, it is mandatory to make a simpli ication, a pattern of the objects and phenomena, since these cannot be contained in their complexity and they can neither be exactly expressed in calculation formulae from the quantity or quality point of view. Such a pattern is known as modelling and consists in considering elements and properties insigni icant in the evolution of the phenomena taken for study. Only the dominant properties are retained. Classical mechanics studies one of the simplest forms of motion of the matter, known as mechanical motion; this is de ined as the relation modi ication of the position of a body, in relation to another body taken as reference. The systems we take for studying, from the point of view of their mechanical motion and made up of one or several bodies, bear the name of mechanical systems. In classical mechanics that varies in time, that is if passing from one "set" of values to another, we say that the system passes from one condition to another, or that an evolution process occurs. Assemblage , that we wish to look into, conveys the relation between the actions exerted on a system and its transformations, that consequently occur, bears the name of dynamic system, understanding that these conditions refer, irst of all, to a mechanical system to which the dynamic system is associated. Any real dynamic system, be it physical, chemical, biological, economic etc, is de ined by certain properties of structure, shape and response to outside actions. It is characterized by assemblage M of conditions that it could have at any moment in time t, belonging to a certain interval I, which in general changes with the lapse of time. Each condition q of the system, n every moment tÎI, is characterized by

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an independent and complete system of numerical parameters (qi), called condition parameters, generally derived from observation and measurement. Assemblage M of all possible conditions of a dynamic system bears the name of space of conditions of that system, or space of con iguration. If the condition parameters vary in time, that is pass from one “set” of values to another, we say that the system passes from one condition to another, or that an evolution process occurs. Assemblage bears the name of space of evolution. We shall say that an evolution process is determinist if its entire evolution (history), that is both its past and its future, is uniquely determined by the current condition. An evolution process is called inite dimensional if the space of its conditions is locally Euclidian in inite dimension, in other words, the number of parameters necessary to the local description of the condition of the system is inite. The process is called differentiable, if the space of its conditions has a structure of differentiable variety, and the changes of condition in time can be described by differential functions. Next, we shall consider only determinist, inite dimensional and differentiable dynamic systems (with evolution). To achieve the target aimed, the following problems must be taken into consideration: a) to build a mathematical model corresponding to a given system. b) to study such a built model, in other words, to high lighten its properties. c) to select the main or dominant actions governing the phenomena considered. d) to simulate the behaviour of the system being studied. Our work program refers only to the irst two problems. The mathematical models that we use in the study of mechanics, are ideal models and contain systems of notions and axioma (postulates) that characterize from the quantity and quality point of view the mechanical, physical, chemical, biological or economic or nature phenomena. Classical mechanics is thus a mathematical model of the “world” of phenomena in motion. As mentioned above, when studying mechanical systems and more exactly dynamic systems, we use several formalisms, such as: Newtonian, LaGrange, Hamiltonian, Routhier, Poissonian, Jacobian or Brinkhoff. In this book we aim to present the most important of these formalisms and to make a comparison among them, with the purpose of highlighting their generality. We shall also present the ideas that lead to expressing and solving the “inverse problem of Newtonian mechanics”, ideas via which it was suggested and inally expressed a new law of mechanics (actually of the nature having applications to the formalisms of mechanics), marking also a few of the main moments of the history of demonstrating the validity of this law. We emphasize the equality between the causalist–determinist point of view of classical mechanics, shaped up by the Newtonian formalism, and the inalist point of view, de ined within the LaGrange formalism, or as it is also known, that of analytical mechanics.

DDS I CONTENT PART I DYNAMICS OF SHAPELESS MATTER

Chap. I

THE CAUSALIST, DETERMINIST PRINCIPLE . THE NEWTONIAN FORMALISM OF CLASSICAL MECHANICS

Page

§ 1.0

Dynamics of shapeless matter

13

§ 1.1

Conventions and notations

15

§ 1.2

Classical dynamic systems and their causalist evolution

23

§ 1.3

Newton’s theory of gravitation

29

§ 1.4

Critics of Newton’s mechanics

36

Chap. II

THE FINALIST PRINCIPLE, THE LAGRANGE FORMALISM OF CLASSICAL MECHANICS

40

§ 2.0

Introduction (history)

40

§ 2.1

The definition and properties of the LaGrange function

45

§ 2.2

The fundamental problems of variational calculus

48

§ 2.3

The inverse problem of classical mechanics

58

§ 2.4

The assemblage of lagrangians of a dynamic system

74

§ 2.5

The LaGrange dynamics of continuous media

84

§ 2.6

Variational systems, auto-adjoint second-degree dynamic systems

93

§ 2.7

The auto-adjucntion of differential equation systems written in the Euler – LaGrange formalism

106

§ 2.8

Second-degree differential equations written in main form

113

§ 2.9

The LaGrange formalism of continuous media

117

Chap III CAP. III THE HAMILTONIAN FORMALISM OF CLASSICLA MECHANICS § 3.0

Introduction

128 128

9

§ 3.1

Autonomous Hamiltonian systems

129

§ 3.2

Non-autonomous Hamiltonian systems

142

§ 3.3

Non-Hamiltonian dynamic systems

158

§ 3.4

The Hamilton formalism of continuous media

167

CHAP. IV

THE

HAMILTON – JACOBI FORMALISM

172

§ 4.1

The Hamilton – Jacobi equations

172

§ 4.2

Action as a time function

174

§ 4.3

The LaGrange variant

176

CAP. V

THE RAUTHIAN FORMALISM

177

§ 5.1

Definition of Routh function

177

§ 5.2

Applications

181

§ 5.3

The LaGrange function of a routhian dynamic system

182

CHAP. VI

THE BIRKHOFF FORMALISM OF CLASSICAL MECHANICS

185

§ 6.1

The Birkhoff dynamics

185

§ 6.2

Birkhoffian dynamic systems

190

§ 6.3

The inverse problem of classicla mechanics in the Birkhoff formalism

194

CHAP.VII

§ 7.0

THE CONSERVATIVE PRINCIPLE, THE NEUTHERIAN FORMALISM 214

PRIME INTEGRALS OF A FIRST-DEGREE ORDINARY DIFFERENTIAL EQUATION SYSTEM

214

§ 7.1

CONSERVATION LAWS OF THE NEWTONAIN FORMALISM

217

§ 7.2

CONSERVATION LAWS OF THE LAGRANGE FORMALISM

220

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§ 7.3

CONSERVATION LAWS OF THE HAMILTONIAN FORMALISM

237

§ 7.4

SYMMETRY TRANSFORMATIONS AND NOETHER THEOREM FOR DISCRETE SYSTEMS

255

§ 7.5

SYMETRY TRANSFORMATIONS AND NOETHER THEOREM FOR DEFORMATION CONTINUOUS MEDIA

271

§ 7.6

THE APPLICATION OF MOMENTUM AND CONSERVATION LAWS

276

§ 7.7

CANANOID TRANSFORMATIONS AND CONSERVATION LAWS

279

PART TWO DYNAMICS OF BIOLOGICAL AND ECONOMIC PROCESSES. 286 Chap.VIII

THE LAGRANGE FORMALISM ASSOCIATED TO BIODYNAMIC SYSTESM

288

§ 8.1

Biodynamic systems with an even number of condition parameters

288

§ 8.2

Biodynamic systems with an odd number of condition parameters

308

§ 8.3

The general case of biodynamic systems independent of the evenness of the condition 316 parameters number

§ 8.4

The LaGrange formalism associated to certain biodynamic systems

319

§ 8.5

Biodynamic systems on Riemann spaces

333

Chap. IX

STUDY METHODS OF BIODYNAMIC SYSTEMS

341

§ 9.1

A system study program

§ 9.2

Behaviour of biodynamic systems to diffeomorphisms in general

344

§ 9.3

Pseudogroups of one-parameter transformations

355

Chap. X

§ 10.1

341

EXAMPLES OF BIODYNAMIC SYSTEMS

The neuronal dynamics

368

368 11

§ 10.2

The epidemics evolution dynamics

377

§ 10.3

Economic processes dynamics

393

CHAP .

OPTIMAL DYNAMIC SYSTEMS

408

§ 11.1

General problems of optimization of certain dynamic systems

408

§ 11.2

Optimal control dynamic systems

421

CHAP. XII

SPECIAL PROBLEMS OF OPTIMAL CONTROL DYNAMIC SYSTEMS 433

§ 12.1

The Hamilton expression of control dynamic systems

433

§ 12.2

Optimal control dynamic systems, the LaGrange expression

442

Biography

476

XI

12

DYNAMIC AND DIFFERENTIAL SYSTEMS, PRINCIPLES AND FORMALISMS Il libro della natura e scritto nella lingua di matematica Galileo Galilei



CHAP. I CAUSALIST – DETERMINISTIC PRINCIPLE. CAUSE AND EFFECT NEWTONIAN FORMALISM BRIEF HISTORY OF THE EVOLUTION OF FOUNDAMENTAL PRINCIPLES OF CLASSICAL MECHANICS It is common knowledge, that of all nature sciences, mechanics is the oldest and second fully axiomatized science (next to geometry).Statically, seen as a irst part of mechanics, it was developed ever since the Antiquity, for instance, the study of the pulley was done by Archytas of Tarent, 400 years before Christ, and its fundamental notions were shaped by Archimedes (287-212) B.C. who also introduced the notion of the moment.In the 17th century, statistics is deemed conclusively founded. The irst scientist who stipulated that statistics is a particular case of dynamics was d’Alembert (1717-1787).Dynamics had a much slower evolution. Its roots can be found in a signi icant collection of practical applications. This is a relatively modern science; its principles being enunciated only in the 17th – 18th centuries A.D. We quote here several illustrious spirits that left a mark in human thinking with concern to natural phenomena and laws by which these apply. They are Leonardo da Vinci (1452-1919), Copernicus (1473-1543), Kepler (1571-1630), Galileo-Galilei (1564-1642), Francis Bacon (1561-1628),Kepler, interpreting the remarks made on planet Mars, determined by Tyche de Brohe (1546-1601), deducted the three laws (1609-1618) that bear his name. The decisive role in interpreting these is played by Galileo-Galilei, who lays the basis of new dynamics. The continuation of Galilei’s creation and completion of a magni icent synthesis is carried out and accomplished by Isaac Newton (1643-1727).The fundamental laws of the evolution of bodies, under the action of forces could be enunciated due to mathematical reasoning, materialized by Newton and Leibniz (1646-1716), by their creation of differential and integral calculus (Fluxional Calculus).

§ 1.0 DYNAMICS OF AMORPHOUS MATTER

1.0.1 DIFFERENTIAL DYNAMIC SYSTEMS OF DISCRETE MEDIA

The abstract notion of system, in its current understanding, assumes a lot of objects (elements, or more generally, parts of an assemblage) in luencing one another and on which external forces are exerted. In their turn, these elements exert an outward force. Among the fundamental properties of every system, we retain that of its own transformation, as well as the possibility to be in luenced, by conveniently selecting the actions to be exerted on the system, inwardly. To study nature, and more precisely a part of it, which for us shall represent a system, it is mandatory to simplify, schematize the objects and phenomena, since they cannot be contained in their complexity and neither can they by expressed in quantity and quality by calculus formulae. Such a schematization is known as modelling and it consists in ignoring the elements and properties insigni icant in the evolution of the phenomena under study. Only the dominant properties shall be retained. Classical mechanics studies one of the simplest forms

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of motion of the matter, known as mechanical motion; this is de ined as being the relative modi ication of the position of a body in relation to another body deemed as reference. The systems we study, from the point of view of their mechanical motion, and made up of one or several bodies, go by the name of mechanical systems.With classical mechanics, the condition varies in time, that is if passing from one “set” of values to another, it is called that the system passes from one condition into another, or an evolution process takes place. Assemblage , that we would like to look into hereunder, and carries the relationship between the actions exerted on a system and its transformations that consequently occur, is called a dynamic system, being implied that these conditions refer irst of all to a mechanical system to which a dynamic system is associated. Any real dynamic system, be it physical, chemical, biological, economical etc. is de ined by certain properties of structure, shape and reaction to exterior actions. It is characterized by assemblage M of conditions that it can be in at any given moment in time t, belonging to a certain interval I, and I general, it changes with the time pass. Each condition q of the system, in every moment tÎI, is characterized by an independent and complete system of numerical parameters (qi), called condition parameters, that are generally generated via observation and measurement. Assemblage M of all possible conditions of a dynamic system is called the condition space of that system, or the con iguration space.If the condition parameters vary in time, that is if they pass from one “set” of values to another, it is called that the system passes from one condition to another, or that an evolution process takes place. Assemblage is called an evolution space. We say that an evolution process is deterministic if its entire evolution (history), that is both its past as well a its future, is uniquely determined by its current condition. An evolution process is called initedimensional if the space of its conditions is a inite dimension locally Euclidian, in other words, the number of parameters necessary to locally describe the condition of the system is inite. The process is called differentiable, if the space of its conditions has a differentiable variety of structure, and the changes of condition in time can be described by differential functions. Following, we shall consider only inite-dimensional and differentiable dynamic (with evolution) deterministic systems.In order to reach the target envisaged, it is mandatory to consider the following issues:

a) building of a mathematical model corresponding to a given system.



b) studying such a built-up model, in other words, high lighten its properties.



c) selecting the main, or dominant actions governing the phenomena considered.



d) simulating the behavior of the system under consideration.

Our work’s program refers only to the irst two issues. The mathematical models we use to study mechanics are ideal models and they contain notions and axioms (postulates) that characterize mechanical, physical, chemical, biological or economical natural phenomena from the quantity and quality point of view. Classical mechanics is thus a mathematical model of the “world” of phenomena in motion. As mentioned above, for the study of mechanical systems and more precisely that of dynamic systems, we use several formalisms, such as: Newtonian, Lagrangian, Hamiltonian, Routhian, Poissonian, Jacobian or Birkhof ian.With this book we intend to present the most important of these formalisms and to make a comparison in between, aiming to high lighten their generality. We shall also present the ideas that lead to the emergence and solution “of the reversed problem of Newtonian mechanics”, ideas via which a new law of mechanics was suggested and inally enunciated (actually that of nature with application to the formalisms of mechanics), also marking some of the main moments in the history of demonstration of the validity of this law. It highlights the equivalence between the causalist-deterministic point of view of classical mechanics, characterized by the Newtonian formalism, and the inalist point of view, characterized within the Lagrarian formalism, or as it is better known, that of analytical mechanics. 1.0.2 DIFFERENTIAL DYNAMIC SYSTEMS OF CONTINUOUS DEFORMATION MEDIA To a good extent, the study of ordinary differential equations can also be extended to the partial derivate equations. A geometrical setting of the presentation of the theory of a dynamic system of continuous media imposes taking into consideration some differentiable varieties, endowed with certain structures. The differentiable varieties considered are real inite dimensional, Harsdorf and

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coordinates of ibrate E shall be noted with (xi,ya) cu i= and a= , and the local section assemblage on an open interval of M, by SectE/U={s:M®E½p*s(x)=x, "xÎU}. We shall note with T(E) and T(M) the vector beam tangent to E and respectively M.The reversed problem of dynamics of systems with a inite number of freedom levels is completely solved and nicely presented by Santilli in [I]. The expansion of this problem to the instance of continuous deformation media and especially to that of the ield theory, initiated by [II], was re-iterated by several authors, among whom we can quote: Helmholtz, Tonti, Anderson, Henneaux. (Chap. II)This chapter intends to methodically present, via generalization, the analytical study of dynamics of continuous deformation media, to present techniques of obtaining presentation laws associated to this type of dynamic systems and to study the reverse problem. The principles and methods of analytical mechanics used by us in the previous volumes on studying the evolution of dynamic systems with a inite number of condition parameters, can also be extended to the dynamic systems of continuous deformation media. Taking into account that these systems have a inite number of condition parameters, consequently the differential equations describing the evolution cannot have the form of Euler–Lagrange or Hamilton equations in the chapters dedicated hereto, since their number that coincides with the number of condition parameters would have to be in inite. Thus, it is necessary to determine an Euler–Lagrange or Hamilton type equation system, other than those known and enunciated in the quote, for continuous deformation media. Such equations (with partial derivates) shall also be useful in the study of electromagnetic ield, in the general relativity theory, as well as in quantum mechanics. E. Noether [ ] 1882-1935 showed that when the evolution equations derive from a variational principle (called Hamilton principle), there can be established a systematic procedure of tracking down certain preservation laws, directly from this principle. We shall later present such a situation in the case of continuous deformation media. In the chapter treating the continuous media, the issued raised, the results obtained as calculus methods, copy by generalization their correspondents in the previous chapters. In another volume we shall pay special interest to building up a ield theory. In this sense, to an arbitrary ield we shall associate a super ield and respectively Maxwell type equations. The study of super ields, associated to such systems, is undoubtedly incomplete; in this much more complex situation, there is still a lot to work. Another issue that we intend to generalize is that of also associating geometrical objects and structures to certain such systems.

§ 1.1 CLASSICAL DYNAMIC SYSTEMS AND THEIR CAUSALIST EVOLUTION 1.1.0 GENERAL NOTIONS As it is well known today, any axiomatized science – and mechanics is one such science – ever since its foundation by Isaac Newton in 1686 [38], presents a series of notions, called fundamental,

notions that cannot be de ined by others, but they can still be described by certain properties characteristic only to them. Among the fundamental notions of mechanics, we count: the space, time and mass. By means of undamental notions and their logical relationship, a second class of notions can be deducted, called derivates. The irst among these are: motion, rest, marks, slopes, speed, acceleration, force, impulse, energy, work, power, continuous medium etc.The fundamental properties that they meet are expressed by a system of axioms, as currently called in Physics: “postulates”, properties that cannot be demonstrated; they are assumed to be true apriori. The fundamental notions and postulates to which these are subject allow us to consider mathematical models and thus model the real motion “world”. Among the simplest and systematically met models in the study of mechanics we quote: the material point, discrete and inite systems of material point, continuous media.In this introductory chapter we shall make a brief description of the simplest notions of classical mechanics (§ 1.1), in the causalist concept, as well as the approach from deterministic to causalist (§0.3)Dynamics is the part of mechanics that studies the motion of bodies under the action

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paracompact. The various applications (functions) that occur are supposed to be class C¥.The notion of ibrate de ines a trivial local ibrate p:E®M (noted still, to make it simple, with E), with the base space M, a differentiable variety of dimension m and total space E a variety of dimension m+n. The local

of exterior forces, the relationship between forces and motions. The notion of dynamics was irst used by Leibniz, in the paper called “Specimen dynamicum“, issued in 1695. The word “dynamics“ is of Greek origin: “dunamis“ means force, power. We call a dynamic system a system that ampli ies the motion of its components under the action of certain given forces. These can be: mechanical, acoustic, electric, magnetic, electronic, etc. In his famous paper “Philosophiae naturalis principia matematica“, published in Cambridge on 8th of May 1686, Isaac Newton enunciated under the title force action principle the second fundamental law of mechanics, according to which, in an inertial mark, a mass body m, taken for material point subject to the action of a force F, receives an acceleration a proportional to the force: ma = F. This is the fundamental law of mechanics. Actually, this is the content of a dynamic system notion, which mathematically translates into a system of coordinates, via a system of ordinary differential quadratic equations: ( 1.1.1 )

, Fi = m f i .

The dynamic systems, such as they are met in mechanics, are presented under local form as coordinates, by basically Galilean and on space restrictive ields, wherever the given equations have solutions and can be integrated in order to obtain the laws of motion. This point of view represents the causalist principle in philosophy and science, according to which the evolution of natural systems, that are modelled by the “dynamic systems”, occur as effect due to the existence of the other systems (bodies) in the universe, acting upon the given system, considered as the cause. As a consequence of the causalist principle, by integrating the equations (0.3.1) of dynamics and by deducting the laws of motion: in the solutions of the differential equations considered we can recognize the deterministic principle, according to which knowing the condition of the system at a given time involves knowing it in all previous (the past) and subsequent (the future) moments, thus knowing the evolution of the system. This chapter intends to study such dynamic systems, trying to de ine them globally, on an arbitrary support differentiable variety, without turning to local maps, in whose subsequent translation of results to obtain work formulae, simple for computations. Classical (Newtonian) mechanics has as object the study of a material point (body) motion or, more generally, of a material point system in an Euclidian space of dimension 3. On this space, there acts a group of transformations (rotations and translations) of dimension 6. The main notions and general theorems of Newtonian mechanics, even though they are enunciated in Cartesian coordinates, are invariable to the more general group of Galilean transformations of time-space. Newton’s equations allow the complete studying of a great number of important problems, among which we can quote the evolution in a central ield. As it is well known, a potential mechanical system is de ined if given the masses of the system points and its potential energy, and the isometrics of space that are ixed potential energy have corresponding preservation laws. 1.1.1 GEOMETRICAL STRUCTURE OF TIME-SPACE IN NEWTONIAN FORMALISM 1.1 Newtonian mechanics operates with notions such as: space, time, mass, material point, force, mark etc., that in order to be veri ied are subject to certain laws called postulates. The consequences thereof deducted via logical reasoning represent the content of “classical” or “Newtonian mechanics”. Let’s admit that the “universe” or Galileo - Newton’s time-space is a differentiable variety V4 of class C , the universe points thereof shall also be called events. Its geometric structure possesses, hypothetically, the following topological and differential properties: V4 is diffeomorphic with R4 and endowed with a family F of diffeomorphisms f of class C

on

:

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, we get:

applications postulate).

/R and

varieties

, where

is a constant¸ in other words, the

/R are thus deducted from one another by translation (time universality

The assemblage of reverse images by

of sub-varieties

of

, associated to f, are sub-

of V4 , independent of f.

A (global) map of V4 is called a system of admissible coordinates if it is given by a diffeomorphism F. Two admissible coordinate systems are thus connected via relations such as:

(1.1.1)



Let’s call a) evolution in R3, a differentiable application

, of an interval I of the real axis R;

b) slope in R3 the assemblage of the values of application q¸ c) slope in V4 the image in f -1 of a slope from

graph of q; d) slope in

; e) ield of efforts in R3 , a differentiable function de ined on

the “time-condition space” and with vectorial values. if

, f)

id the canonical projection on the irst factor:

shall call a ield of forces on V4, the function

we .

Classical mechanics sees the world as being made of “material points”; a material point is characterized by its mass m (non-null), occupies in each moment t a geometrical point q; its speed is by de inition:

, and its acceleration: vector

.

Newton’s principle of determinism tells us that the initial condition of a material point (its position and speed at a given moment) uniquely de ines its subsequent evolution. The material point evolution of mass m (positive and invariable number), is directed by the presence of an associated ield of forces as well as by the initial conditions when it is “launched”, at a given time, into this ield of forces. The law we admit in classical mechanics as governing this evolution (the second fundamental law of mechanics), is given, in an admissible coordinate system of variety V4, by the quadratic differential equations, called Newton’s equations:

(1.1.2)



whose solution, in the initially given conditions, de ines the material point evolution and where the ield of forces F we assume, for the moment, as differentially depending, otherwise arbitrarily, on the coordinates speed and time:

.

The fact that the components of ield F occur as functions de ined on replacement of the equations (1.1.2) with the system:

, suggest the

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So that, if

(1.1.3) V4



Proposition 1.1 Equations (1.1.3) are invariant to the admissible coordinate system changes on

Indeed, in another coordinate system (1.1.3) we have:

(1.1.3’)



From (1.1.1) it deducts: , relations telling that the shapes components, their cancellation being invariant.

are “counter-variant”

Taking into account the formulae (1.1.3), it deducts that along the evolution of slopes, (1.1.3’) and that:

(1.1.4)

(modulo



Formulae (1.1.4) render the “transformation law” of the coef icients F i via an admissible coordinate change of shape (1.1.1). These formulae allow us to introduce a relation in the assemblage of admissible coordinate system and we shall say that two coordinate systems f and

are in relation if:

(1.1.5)



Condition (1.1.5) requests that in (1.1.4) we have

(1.1.6)



whichever be vi. System (1.1.6) is equivalent to system:

(1.1.6’)



The relation induced by (1.1.5) is an equality. It thus splits the admissible coordinate system assemblage into disjoint classes. This property is equivalent to the fact that referential changes

(1.1.7)



represent a group.

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Field F is thus too “counter-variant” to the coordinate changes given by group (1.1.7). If

in a coordinate system, in another one, we might have

only if functions

, obviously this happens if and

are solutions of the system:

(1.1.8)



Proposition 1.2 Being given ield , if there is a coordinate system where F=0, then there is an in inity of such systems and namely all the systems equal to the last one, and only these. Indeed, be it

one such system and

is also a solution to (1.1.8), a direct calculus shows us that

. Reciprocally, be it and taking into account that

, replacing in

the derivates of

in relation to q and t

and that cancelling occurs regardless of v, we get : that must be demonstrated. Along the slopes, the formulae

(1.1.8) become

.

The integration of system (1.1.8) comes back to the integration of the system:

The irst postulate of mechanics, or Galileo’s principle, can be enunciated in any of the classes of equality established above. This is a direct consequence of the relation (1.1.5). If ield F is null in a system f, it shall be null in any other system equal to the irst, and based on proposition 1.2, only in these. A body at rest, or in even motion, continues to maintain this condition in any other equal system. The systems where , shall be called “accelerated”, in relation to the irst. Up to now we have no criterion to select any class at random from the classes of equality determined by (1.1.5), as a privileged class. But one such possibility could occur if assuming disposing of a physical criterion o determine a diffeomorphism f of family F. The admissible coordinate system class equal to that determined by this diffeomorphism confers V4 an af ine space structure. The admissible coordinate systems equal to f are called af ine coordinate systems. Let’s endow now an “af ine space” V4 with a metric structure. To achieve this target, let’s consider standard metric R3 on the ield of two times co-variant quadratic symmetrical tensors of components

, metric de ined by

Via diffeomorphism f this structure transposes on V4 .By any class of admissible coordinate equality, a new equality class can be de ined, supplementary requiring: (1.1.9)



In relation to standard metrics.

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Conditions (1.1.5) and (1.1.9) taken together require that in the coordinate system changes we have: . The coordinate change group, given by (1.1.7) and (1.1.10), bear the name of Galileo’s group. Once given the af ine coordinate systems, Galileo’s group determines an equality in between, any class of equality bears the family name of “inertial systems”, a “preferential” family can be chosen by giving the diffeomorphism f. We get: 1.2 Galileo’s principle of relativity There is no criterion (physical) to select a privileged system from its class of preferential systems. Thus, from the hypothesis issued it results that there is a class of preferential coordinate systems, called inertial, characterized by the following two properties: a) The laws of mechanics are the same in all the systems of this class, b) The systems of this class deduct one from the other via a Galileo transformation. Aiming to compare the Newtonian mechanical systems with the Hamiltonian ones, we shall consider the application: mass m.

, associated with each

De inition 1.1.3 We shall call a “Newtonian” dynamic system N, a “ ield of efforts”: , associated with each mass m, so that the evolution of a material system of mass m, located in this ield (launched into a point), at a given time, with a certain speed, be described by Newton’s equations:

(1.1.10)



1.3 Referential exchanges An event x or “time-space point” is marked by the linear pair , representing the place also be represented by another pair considering two reference

and

of the event and its date

. Yet, the same event can

. We call referential a bijective application

. If

we get

The referential change is translated by the formula . In classical mechanics, it is considered possible to select once and for good the length and time units and to orientate space and time. We saw that mathematically; this expresses by the fact that the passing formulae from the irst referential to the second is given by

(1.1.11)



where

.

Reasonings of physical nature oblige us to take into consideration, besides the inertial marks, the marks “in rotation” to the irst ones.

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depend on t and functions From the fact that

,

are differential.

we deduct the identities:

,

,

,



This latest formula can be transcribed as via x,

, by using the vectorial product operator

,

, or in coordinates:

From

it results that there is a vector

, depending on t, de ined by:

(1.1. 12)



This vector is called “instantaneous rotation” of the new referential in relation to the old one, its components being given by: the formula (1.1.11), we ind

, where (i,j,k) is an even permutation of (1,2,3) Reversing

(1.1.11’)



where and ; noting , we get: . 1.4 Exercise speed and acceleration Now let’s high lighten the relations connecting in between the evolution elements expressed in the two marks. It shall be noted with dq/dt=v the point speed in the irst mark, called absolute,

/dt=

the speed of the same point in the second mark, called relative

and analogous to dv/dt absolute acceleration and the formula (1.1.11), leads us to the relation:

/

(1.1.13) where the expression

relative acceleration. A direct calculus using



(1.1.14) Bears the name of “exercise speed”. Deriving the formula (1.1.13) we reach the formula:

,

(1.1.15) where the expression:



(1.1.16) is called “exercise acceleration” and



(1.1.17)

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One such mark is given by the fact that formulae of the type of those above (1.1.11), where A and r

“complementary acceleration” or Coriolis: the formulae (1.1.13) and (1.1.15) give us the relations sought. We cannot say the same thing about the formulae



From the invariation of formula (1.1.13) we deduct:

(1.1.18)



1.5 The dynamics of material point systems Be it given a system of N material points, acted upon by a ield of forces (generally assumed as the action result of several ields). The ield action on each point leads to the equations:

(1.1.19)

,

Equations describing the evolution of the system given by the N material points Na (no connection) located in the geometrical points qa, marked each by a number a, , respectively affected by masses ma , acted upon by a ield of forces Fa (which may also coincide). With the change of notations

j the equations (1.1.19) present as:

(1.1.20)

,



As experience shows, the formulae (1.1,19) that de ine the evolution of a material point system can be applied in the most varied situations. Yet, most often it is dif icult to know the forces Fa, acting on each point of the given system. Because of this, we must make approximations. One of these approximations, very often used in practice, is that of considering the system “with connections”, that is of imposing dependence (often and of time) relations on the coordinates, generally in the form of:

(1.1.21)



in number of 3N+r and satisfying, for instance, the condition;

(h=r+1,....,3N) The formulae (1.1.21) de ine in a dimensional variety represented, from the parametric point of view, by

(1.1.22)

, n=3N-r, that we assume, locally,



In order to rewrite the evolution equations (1.1.20), in coordinates

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, using he formulae

(1.1.23)



we get:



(1.1.24) which are the sought equations.

§ 1.2 CLASSICAL DYNAMIC SYSTEMS AND THEIR DETERMINISTIC EVOLUTION 1.2.1 Differential dynamic systems and their evolution deducted from the deterministic principle [1] Any real dynamic system, be it physical, chemical, biological, economic etc., is de ined by certain properties of structure, form and reaction to outside actions. It is characterized by the assembly M of conditions, that it may have at any moment in time t, belonging to a certain interval I, and that generally changes with the lapse of time. Each condition x of the system, at any moment tÎI, is characterized by a independent and complete system of numerical parameters (xi), called condition parameters, that generally derive from observation and measurement. De inition1.2.1: The assemblage M of all possible conditions of a dynamic system bears the name of condition space (con iguration space). De inition 1.2.2.: If the condition parameters vary in time, that is if they pass from one set of values to another, it is called that the system passes from one condition to another, or that an evolution process occurs. De inition 1.2.3: We shall say that an evolution process is deterministic if its entire “history”, that is both its past and its future, is determined by the current condition. De inition 1.2.4 : An evolution process is called inite dimensional if the space of its conditions is of a inite dimension, that is the number of necessary parameters to describe the condition of the system is inite. De inition 1.2.5.: The process is called differentiable, if the space of its conditions has a differentiable variety structure, and the condition changes in time can be describes by differential functions. Next, we shall consider only deterministic dynamic systems, inite dimensional and differentiable.

We shall note with x0 the condition of the system at a given time t0ÎI and with x1 that corresponding to moment t1ÎI, and passing (evolution) from condition x0 to condition x1 with:

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: x0ÎM®

(x0)=x1ÎM.

In this sub-paragraph it is shown that any deterministic system is causalist. We shall note with x0 the condition of the system at a given time t0ÎI and with x1 that corresponding to moment t1 Î I, and the passing (evolution) from condition q0 to condition q1 de ined by: bears the name of evolution (transformation) operator; this is thus n application de ined by the space of conditions M with values in M. The assemblage of evolution operators is an application family f indexed by two parameters that has to enjoy the property: , from where it results

is the identical transformation of M.

The hypothesis that the evolution process is deterministic imposes the obligation that conditions: and

be equal.

If the initial moment is set as t0ÎI and at the same time the initial condition x0Î M of the evolution process, the assemblage of conditions x of the system represents, within the condition space, a family of conditions dependent on one single parameter t that is a curve, called orbit or slope. De inition 1.2.1 Be it S = (M, I, f ) a triplet where:

- M is a differentiable variety of inite dimension m, arbitrary, called the space of conditions,



-I Ì R is an interval of the real axis and



-f is a application (family of applications, indexed by I ´ I):

f : (t, r) Î I ´ I ® f(t , r) Î { f : M ® M}, where { f: M ® M} is the assemblage of the applications of M in M. The triplet S is called deterministic evolutive system, inite dimensional and differentiable, and F the evolution process, if:

f(t , t) = idM , " tÎI,

(1.2.2)

f(t , u) . f(u , r) = f(t , r) , " t , u , rÎI

the formulae (1.2.2) represent what is called the Champman – Kolmogorov Law Observation From the properties (1.2.2) it results:

f(t , r) . f(r , t) = f(t , t) = idM f(r , t) . f(t , r) = f(r , r) = idM

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which tell us that f(t , r) = f-1 (r , t) that is functions f(t , r) are bijections, " t, rÎI. On the Cartesian product I ´ I, a there can be enunciated a composition law partly de ined by: (t , u) (v , r) = (t , r), " t , u , r ÎI if and only if v = u. The composition operation thus de ined is, obviously, associative and with null effect elements in the form of (t, t) (diagonal) and the reverse of any element (t,u) is element (u,t). Consequently, the assemblage I´ I endowed with such a composition law represents a pseudo-group. If f is a deterministic evolution process, then application (t,r)® f(t , r) is a morphism of pseudo-groups, from the pseudo-group I´ I to the pseudo-group of evolution operators, since it veri ies the relation: f[(t , u),(u,r)] = f(t , r) = f(t , u) f(u , r) De inition 1.2.2 Be it S a deterministic evolutive system, and x0 Î M an arbitrary point set in the space of conditions and t0 Î I set. Let’s consider application j :I ® M, de ined by:

(1.2.3)



It is said that the evolution of the system that at moment t0 is in condition x0, the application (1.2.3) of segment IÌ R in the space of conditions M, and the application image j bears the name of slope or orbit of x0 . A point x0, with the property:

= x0 , " t ÎI, is called a balance point of the process. .

Generally, the evolutive systems and processes described by function , depending on two timevariables are called non-stationary evolutive processes. These can also be described via function depending on one single time variable, as following: Be it S a deterministic evolutive dynamic system and a ÎI a set real number; Then, whichever be " t ,r ÎI we get: f(t , r) = f(t , a) f(a , r) = f(t , a) f-1 (r , a) , such as, if we note f(t , a) = ja(t) = j (t), we get a family: {ja(t),tÎI}= {j(t) }tÎI of bijections of M, with the property:

(1.2.4)

f(t , r) = ja(t) j-1a(r) = j (t) j-1 (r), " t ,r ÎI.

It results that a deterministic evolutive process f de ines with each aÎI, one family {ja (t) }tÎI , dependent on parameter t , on the bijections of M and each such family determines process f by the relation (1.2.4). If bÎI, with b ¹ a , then {jb (t) } ¹ {ja (t) }. Yet: j b (t) . j-1 b (t0) = ja(t) j-1a(t0) =



Reciprocally, any family of bijections {j(t) }tÎI of M, de ines on M a deterministic evolutive process via relation (1.2.4). Any two families of bijections of M: {j’(t) }tÎI and {j”(t) }tÎI are called equal if:

(1.2.5)

j'(t). j'-1(r) = j"(t). j"-1(r), " t, r ÎI,

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Relation (1.2.5) is re lexive, symmetrical and transitory, thus it is a relation of equality in the assemblage of families of bijections of M, depending on one parameter tÎI. each class of equality de ines a deterministic and reciprocal evolutive process. Evolution equations Be it S = (M, I, f) a deterministic and non-stationary evolutive system and the differentiable application:

(t, t0, x0) ÎI´I´ M ®

, x0 ÎM.

For any set t0 ÎI and x0 Î M, application tÎ I ®

[

x0 Î M is differentiable, thus there is a limit:

x0 – x0 ] = [

x0 ]t=t

We note this limit with X (t0, x0). For each set t0 ÎI, application: x0 ÎM ® X (t0 , x0 ) ÎTM, is a differential operator de ined on M. the operator family X (t, x) is called the evanescent generator of the non-stationary and non-autonomous evolutive process. Theorem 1.2.3 Any evolution t ® the problem of Cauchy:



x0 = j(t) of a non-stationary evolutive process f is the solution to

(1.2.6)

, j(t0) = x0

Demonstration j(t) =

x0 then, from the de inition of the derivate. We get:

as well as j(t0) =

x0 = x0, by expression, j(t) is the solution to the problem of Cauchy (1.2.6).

It results that a non-stationary and non-autonomous deterministic evolutive process generates on M a family of operators {X(t,x)} so that any orbit j(t) = differential equation:



x0 of the process meets the non-autonomous

(1.2.7)

.

This equation represents the local evolution law of process f, allowing to recapture the evolutive process f when the evanescent generator of the process is given. This result is important in the sense that it tells us that the evolution of a system can be more easily expressed via the evanescent generator of the process than by the description of the process itself. Consequently, the evolution laws of various systems are gotten in differential form, and in order to get the evolutive process, it is necessary to integrate this differential equation (1.2.7).Not any family of operators {X (t, x)} = {Xt} differentiable on I

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´M is the evanescent generator of a deterministic and non-stationary evolutive process. The theorem of existence and uniqueness for ordinary differential equations ensures the exist ace and uniqueness of the problem of Cauchy (1.2.6) only locally, for a suf iciently small vicinity of t0, | t – t0| £ e.There are examples where the local solution to this problem is not extensible to the entire time interval I. The family of operators Xt above considered determines an evolution law only in the case when the solution is extensible to I.Yet, a family of operators Xt differentiable on M de ines in the vicinity of point (t0, x0)ÎI ´M, what we call a local family of transformations

of M.

De inition 1.2.4 We call a local family of transformations of M, de ined by operators Xt , in the vicinity of point (t0 ,x0 ), a triplet (I’,V0 ,f), made up of an interval I’ ={tÎ R ; | t – t0| £ e}, a vicinity of V0 of point x0Î M and an application f : I’ ´ I’ ´ V0 ® U Ì I ´ M , so that: 1. For any set t, t0 Î I’, application f , de ined by:

: I’ ´ I’ ´ V0 ® U is a diffeomorphism.

2. For any set point (t0, x0) Î I’´ V0, application j (t), de ined by 3. (t) =

, is the solution to the problem of Cauchy (1.2.6).

4. There occurs the property = şi = x0, for any t0 , x0 , t1 , t, for which the right member of the equality is de ined. Besides, for any x0 Î V0 there is a vicinity V and d > 0 , so as the right member be de ined for any x’0 Î V’ and | t – t0| £ d . On a local map, with the coordinates x0 = ( x1(t0) ,…, xm(t0)), we have: j (t) =

= ( x1(t0) ,…, xm(t0)).

Each evolution of the process is a curve de ined by functions xi (t), and the vectorial function (t,x) ® X(t,x) is expressed by functions Xi(t, xh ). There results that the evanescent generator associates to each point xÎ M, a vector X(t,x )= Xt (x) Î Tx M, variable in time, generating on M a non-stationary ield X(t,x). Each orbit j (t) = (Xi (t)) of the process meets on the respective map the non-autonomous system:

with coef icients Xi(t,xh) equa to the components of ield Xt (x). A special yet very important case for the applications is that of the autonomous deterministic evolutive systems. De inition 1.2.5 Be it S an evolutive system; S is called an autonomous evolutive system, and f an autonomous evolutive process, if: 1. f is deterministic evolutive, that is f : I x I ® F(M) meets the conditions: a) f (t ,t) = idM , " t Î I , ( 0 Î I ), b) f (t ,u) . f (u ,r) = f (t , r) , " t , rÎ I, 2. f is autonomous if: , " t , t0 , h Î I.. Condition 2 can also be expressed by: f (t +h ,h) = f (t ,0), " t , h Î I cu t + h Î I,

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or: f (t ,r) = f (t-r ,0), " t , rÎ I

and t -rÎ I.

A determinist and autonomous evolutive system does not depend on the two parameters t and r, but only by their difference, reason for which it was called autonomous. Theorem 1.2.6 Be it S a deterministic and autonomous evolutive system and f (t ,0) = j (t) then:

(1.2.8)



Demonstration According to the notations, it is written as:





j (t) . j (r) = j (t + r ), " t , r , t + r Î I.

j (t) . j (r) = f (t ,0) f (r ,0) = f (t+r ,r) f (r ,0) = f (t+r ,0) = j (t +r).

Relation (1.2.8) shall say if I is deemed an additive pseudo-group, then application: tÎI ® j (t) Î F(M) = group of bijections of M, is a morphism of pseudo-groups. From condition (1.2.2) it results that f(t, r) are bijections of M, f (t, 0) = j(t) and are also bijections of M. Be it S = {M, I,F} a deterministic evolutive system, differentiable and autonomous. Then:

We see that whatever be set t0 Î I, application x0 ÎM ® X(x0) ÎTM is a ield of vectors on M, described on a local map by Xi( xh), this is now an autonomous ield and it is called still evanescent generator of the process. Theorem 1.2.7 Any orbit j (t) = problem of Cauchy:

, of an autonomous evolutive process is the solution to the

, j (t0) = X0

In the demonstration we shall also take into account the condition of autonomy, thus:

and j (t0) =

on a local map, with the coordinates xi, Cauchy’s problem can also be written as:



, xi(t0 ) = xi0.

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28

f



, c.c.t.d

There results that an autonomous evolutive process generates on M a ield of vectors X(x) that does not explicitly depend on time and so that any orbit j(t) = (xi(t)) of the process is the solution to the autonomous system:

, whose coef icients Xi(xh) are given by the components of the evanescent generator X(x). Application j:I´V® M, noted with {j(t)}tÎI is called local current in the vicinity of x, of the ield of vectors X(x). Family {j:I´V®M}, of all local currents associated to X(x), is a pseudo-group with a parameter of local diffeomorphisms generated by ield X(x). To conclude with, if we consider now that variety M the variety tangent to , of a variety of con iguration , and on it an evolution process, there results the property that the evolution curves (system’s slopes) are the solutions of a quadratic differential equation system, which proves that any deterministic system is causalitic as well.

§ 1.3 NEWTONIAN THEORY OF GRAVITY [C I]

In 1609 and 1618 Kepler enunciated the laws that go by his name. from these laws, it results that: 1. the planets, assimilated with material points, describe curves and planes and certain ellipses around the sun, with the sun being located in one of their ixed points, 2. the motion goes by the law of areas, 3. the relation between the cube of the semi-major axes of the ellipse: a3 and the square of the evolution time: T2, is the same for all planets. From these laws, Newton deducted the law of the force of attraction that the sun exerts on the planets. As it is known, the law of gravity, enunciated by Newton, says, as a irst principle, that every object in the universe, attracts (interacts) another object with a force that is proportional to their respective mass and varies reversely proportional to the square of the distance between them. Mathematically, this statement is expressed by the formula:

, As a second principle, an object reacts to a force (caused by another object) suffering (as an effect) an acceleration in the direction of the force, and (of a contrary direction) reversely proportional to its mass. The law of universal attraction is rigorously veri ied by the experience of Cavendish. It explains the modality in which the bodies move in the Universe but doesn’t say anything about what or who makes such a motion to occur. 1.3.1 GRAVITATIONAL FIELD If one of the bodies is a large body, for instance a star like the sun, planets, the moon etc., ad the second is a small one, deemed of a mass equal to 1, and placed into an arbitrary point in space (deemed threedimensional), then in that point of the space a vector F gets associated (in relation to a class of systems

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of very special coordinates), and it is said that the totality of these vectors represents a ield, expressed by the components: , where (xi) are the coordinates of the test point, and

those of the attrition center.

1.3.2 THE NEWTONIAN POTENTIAL OF GRAVITATIONAL ATTRACTION In the quasi-totality of the books on mechanics, we come across chapters about the mechanics of the material point. In reality, there are no such points. Yet, the hypothesis of the existence and studying of the material points is fully justi ied by the behavior, in certain reasoning, of macroscopic solid bodies. Celestial mechanics assimilates the sun and the planets with material points. Justifying such an assimilation is done in the theory of Newtonian whose fundamental elements were laid by Newton.According to the law of Newtonian gravity, the attraction force it exerts on a material point , of mass m, on another material point P, of unity mass, is:

(1.3.1)

,

with

. The ield of forces F is a conservative ield. (1.3.1).

The projections of force F on the coordinate axis are partial derivates of a function:

and may be interpreted as

, where

(1.3.2)

, such as: .

The projection of F on an arbitrary direction of unit normal If there are more attraction centers Qj , ( attraction force manifesting on point P, is:

, where

is

.

), respectively of masses mj , then the expression of the

.

Laying down , we get (1.3.2). Function U(P) is the Newtonian potential corresponding to the material point system Qj. If we assume now a continuous medium in the form of: a) an attractive material curve c, the expression of Newtonian potential distributed on c, is:

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,

,

where ds is the linear element and

is called speci ic linear mass

b) an attraction medium distributed on a surface S, then:

is the surface element and where

bears the name of super icial speci ic mass.

c) if the attraction matter is distributed in a volume V, then the expression of Newtonian potential is:

is the volume speci ic mass and

the volume element.

Exemple:1o The surface potential of an homogenous sphere The plain layer potential is the sphere is given by:

, being here a constant, and S is the equation surface . Distance is easily evaluated. For symmetrical reasons, function U shall only depend on the distance of the attracted point P to the center of the sphere. We can assume that we take axis Oz so tat is passes through point P, this only to simplify calculations. Be it P = P(0.0.z). We have

=

, from where

the co-latitude of point Q, that is angle (Oz,

, noting with θ

).

Taking as curvy-linear coordinates on the sphere, length φ and co-latitude θ, we shall obtain:

. The irst integral is calculated if taking for numerator the expression differential beneath the root sign and we obtain:

. Thus:

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, if z >R and

, if -R0, c,k¹0. With the denotations

the equation (3.1.20) is equivalent to the irst order system:

The vector ield X de ined on R2 cannot be associated with any symplectic structure in any neighborhood of the origin. Indeed, looking for an integrating factor regardless of time, solution to the equation (3.1.20), we are led to the request that impossible to accomplish because chosen:

The integrating factor us determined on an open conveniently

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where

The X ield id Hamiltonian local related to the 2-form , and Hamilton’s corresponding function is

The property of a ield of being Hamiltonian local is expressed by: Theorem 3.1.46 A condition necessary and suf icient so that a vector ield X to be Hamiltonian local is that:

(3.1.22) Proof. We will take into account the relationship (3.1.4) and we have: LX({f,g}) = LX(w(Xf,Xg)) = LXw(Xf,Xg) +w(LXXf), Xg)+w(Xf, LX(Xg))=

= LXw(Xf,Xg) +w([X,Xf], Xg)+w(Xf,[X,Xg]) = LXw(Xf,Xg) ([X,Xf]) + ([X,Xg]) = LXw(Xf,Xg)-dg([X,Xf]) + df(,[X,Xg]) = LXw(Xf,Xg) – [X,Xf](g) +[X,Xg](f) = = LXw(Xf,Xg) – X(Xf(g)) + Xf(X(g)) + X(Xg(f)) –Xg(X(f)) = LXw(Xf,Xg)+X(w(Xf,Xg))+X(w(Xf,Xg))+d(X(g))(Xf)-d(X(f))(Xg)= =LXw(Xf,Xg)+2LXw(Xf,Xg)+

w(Xf) -

w(Xg)=

=LXw(Xf,Xg) +2LXw(Xf,Xg)+ =LXw(Xf,Xg)+2LXw(Xf,Xg)+{X(g),f} -{X(f),g}. Therefore, we can write: LXw(Xf,Xg) = X({f,g}) = - LXw(Xf,Xg)+{X(g),f} -{X(f),g}, relationship from where the statement results. Observation 3.1.47 The formula (3.1.22) has the analytical signi icance of the derivation related to the Poisson Bracket considered as a product, (3.1.23) The behavior of the Lie algebra Xl(M) to the action of the diffeomorphisms is described by:

Statement 3.1.48 Given

the ield

a diffeomorphism de ined on the symplectic variety M. Then

Xl(M) is Hamiltonian local related to

if and only if

is Hamiltonian local related to

If as well is a simplectomorphism then the image through ield is a Hamiltonian local ield . Proof. The statement is a consequence of the identities (3.1.2).

of any Hamiltonian local



§ 3.2. NONAUTONOMOUS HAMILTO NIAN SYSTEMS



3.2.1 The geometrical structure of nonautonomous Hamiltonian systems

As we have seen, symplectic geometry has proven to be the appropriate geometrical frame to describe autonomous Hamiltonian systems (independent of time). Some of the previous considerations transpose without dif iculty for the case of the nonautonomous systems, others, however, do not generalize immediately. The main difference is the geometrical signi icance of time; in the irst case it appears as a parameter of the integral curves of the vector ield describing the dynamics, whereas in the second case time is a new variable, and the curves obtained by a reparameterization must be considered as being equivalent. De inition 3.2.1 Given V a differentiable variety of size It is called an almost contract structure on V a pair

where

and

is a volume form on V. The

are chosen in such a way that triple is called almost contact variety.

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Statement 3.2.2 If

is an almost contact variety, then there is a unique vector ield

that checks the properties:

(3.2.1)



The R ield is called Reeb’s ield of the almost contract structure . Proof In order to prove the uniqueness of the R ield, let us consider the equation system:

(3.2.2)





Given X and Y two of its solutions; because

is a volume form, from

it

results that X = Y. To prove its existence, it is enough to provide a local demonstration. Because does not cancel itself on V it results that is of 2n rank in any point and therefore is of dimension one for any to each

a vector ield

that does not cancel itself and thus

we will mandatorily have by

for any

Because

Therefore, the vector ield de ined

is a local solution of the system (3.2.1). Observation 3.2.3 Reeb’s ield R does not cancel itself on V and as such de ines a D foliation of one

dimension called the dynamic lux of the triplet hyperplan of

Thus there is on an open U vicinity

. In each point

there is a

transverse to D and the restriction to this hyperplan of the

Example 3.2.4 Given

2-form is of 2n rank.

a symplectic variety. On the variety V = R´M the 1–form

can be de ined, where t represents the standard coordinate of R, and the 2-form are the canonical projections). Then Reeb ield is

Because

Statement 3.2.5 Given

and

is an almost contract structure on V, whose the structure

is also symplectic.

an almost contact variety. The application

(3.2.3) ♭ ,♭ is a vector ibrate isomorphism.

♭



Proof The application ♭ is a vector bundle morphism related to the natural projections on V. It is

therefore enough to show that it de ines an isomorphism from we have the decomposition

collinear with

we would have

for any We therefore have

and

Indeed, if

therefore in the end

’s restriction to Ker

Observation 3.2.6 The isomorphism

♭

induces a vector spaces isomorphism

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♭

is nondegenerate, it is inferred

denoted with ♭ ♭ particularly ♭(R) = q. As it can be seen, statement 3.2.5 plays a central role in the following.

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If



which is impossible. As

on

where

the right member is null only if

that

(p1:V®R and



Statement 3.2.7 Let us consider a V variety and a

couple where

and

so that the ♭ application de ined by (3.2.3) is a vector bundle morphism. Then, one and only one of the following statements is true:

a) V is of even dimension and

is an almost symplectic variety (meaning dimKer



b) V is of odd dimension and



Proof Given R the vector ield with the property ♭(R) = q. We then have



is an almost contact variety.

relationship that leads to

from where it results that the

function may take the

values 0 or 1. Let us now assume that the V variety is connected. If

we have for any

♭

the isomorphism

=

application that V is of 2n dimension and therefore that

If

♭

we have

given

is nondegenerate. Therefore

this is a hyperplan of

(because

is therefore injective. As subspace annihilator generated through

. The

is then surjective, therefore bijective. This proves is of 2n rank.

does not cancel itself on V. For any

and if

(because

The application

this application has its values in the

transversal through

is of even dimension 2n and

It results that

’s restriction to

is of 2n rank. In the end, V is of 2n

+ 1 dimension and is a volume form on V. If V is not connected, the above reasoning is applied on each connected component, and as the dimension of each component is the same, the proof is concluded.

so that the ♭ application

Consequence 3.2.8 Given on a V variety a pair

given by (3.5.3) is an isomorphism. If there is a the triple

ield that satis ies the properties (3.5.48), then

is an almost contact variety.



Proof Because Ker

this excludes the case a) from the previous statement.



Given



(3.2.4)



Statement 3.2.9 The application is an injection. Proof Indeed, if X and Y are two solutions of the system:

an almost contact variety and let us consider the application:

(3.2.4) then

and

However,

= 0,

which implies X = Y, because is a volume form. We have seen that a form system (3.2.4) admits at most one solution. We now state the question regarding the existence of such solution. Given Ker

and Ker

As we have seen in the proof of statement 3.2.4, Ker is a dimension one module generated by Reeb’s ield. We have: Lemma 3.2.10 The necessary and suf icient condition so that the second equation in (3.2.4) to admit the solution is that Proof Indeed, if

Ker



is given then there is a If

ield

so that

using the relationships (3.2.1), we get

Reciprocally, if

for a ield

, one obtains

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f



meaning Ker

♭

from where because





Theorem 3.2.11 If Ker

Ker



Proof Given the equation

then the system (3.5.4) admits a unique and reciprocal solution. and

solution for this equation is

one of its particular solutions. The general

R. Let us now search for a particular solution that also

veri ies the equation

We have:

from where we obtain

The

sought solution for the system (3.2.4) is The uniqueness of the solution is a consequence of statement 3.2.9. Everywhere throughout the rest of this section, we will assume that the almost contract structure

is cosymplectic, meaning

and

It is known that in this case there is

in the neighborhood of any

point a local coordinates system called canonical

so that

We then have



and



De inition 3.2.12 We shall call the Hamiltonian of the function

the vector ield

de ined by

(3.2.5)

is the reverse of the ♭ isomorphism from the previous observation.

where

Observation 3.2.13 The

ield is the sole solution of the system:



(3.2.6) Let us also observe that:



(3.2.7)



In canonical coordinates the local expression of the



(3.2.8)



Statement 3.2.14 Given

ield is:

a 1–form relatively closed related to

Then the couple for any function

where

the

(meaning

de ines a cosymplectic structure on V. Also,

Hamiltonian associated to f for this new structure, coincides with



Proof We have

Reeb’s ield

and

therefore

of this structure is de ined by where

have:

and

veri ies the relationships

[a(R)+

If

other hand

is a cosymplectic variety. From this it results that

and then

Ya(f)

We also

df = df - [ R( f)

On the





Therefore we have



We will now consider a symplectic variety

product variety,

and, because R,

the statement is proven. of 2n dimension and given V:=R´M the

the canonical projections and let us denote with t R’s natural

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coordinate. As we have seen in example 3.2.4, on the variety V a cosymplectic structure (V,

is

naturally de ined, whose Reeb ield is De inition 3.2.15 It is called a nonautonomous dynamic system a ield of vectors dependent on a parameter X:R the relationships:



to the X ield we will associate a

vector ield uniquely de ined by

(3.2.9)



and called the extension of the X ield. For a ixed t, we will denote with vector ield de ined by



Given now a function

The Hamiltonian vector ield

related to the cosymplectic structure



associated to H

is uniquely de ined (according to (3.5.6)) by

(3.2.10)

If

the ("genuine")



represents a Darboux coordinates’ system on the M variety and

corresponding system on V, then

(3.2.11)



The

where

has the local expression:

ield is actually the horizontal lift of the time dependent vector ield is here the symplectic gradient of the H function:

coordinates

R. In the canonical

has the same expression (3.2.11) as

mechanical Hamiltonian system canonical equations:

are the curves

. The V trajectories of the de ined by the Hamilton’s

(3.2.12)



On the other hand, these are the integral curves of the (3.2.13) characterized by:



the

vector ield

(3.2.14)



Let us note that R is actually the extension to V of the time dependent ield The R ield represents of the V variety (of odd dimension) the similar of the Hamiltonian vector ield on a symplectic variety. Because de ined on V:

and

this leads to the insertion of the Lagrange 2-form

(3.2.15)



so that on the V a new cosymplectic structure

is de ined whose Reeb ield is R given by (3.2.13).

According to statement 3.2.14, the cosymplectic structure

admits the same Hamiltonian

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vector ields as the structure structure

particularly

is the Hamiltonian ield associated to H in the

. By construction, the trajectories of the mechanical system are the integral curves

of the Reeb ield R of the structure

, meaning the trajectories of the dynamic lux of

The cosymplectic variety

.

is called associated to the Hamiltonian mechanical system

One can notice that several mechanical systems which have the same con iguration variety may admit the same associated cosymplectic variety. We can therefore speak about equivalent mechanical systems. The product structure V=R and the natural map of R have allowed for the de inition of the vector ield related to

that constitutes a base for (meaning

the module of the vector ields

vertical

Similarly, the dt 1-form de ines a 2n–dimensional distribution

and a vector ield belonging to the distribution is a horizontal ield called semibasic related to

if

A

1-form is

’s contraction with an arbitrary vertical ield is zero, meaning

We will denote with



(3.2.16)



the module of horizontal ields and the semibasic 1-forms, respectively. Given an arbitrary system of local coordinates (on the M variety) the local expressions for horizontal ields, respectively semibasic 1-forms are:



(3.2.17)



Given now an arbitrary 1-form meaning:

We will denote with

(3.2.18)

the semibasic part of



If then the 1-form de ined by:

semibasic differential (or the horizontal differential) is a semibasic



(3.2.19)





These properties can be extended also so higher order forms. Given a k–form

is

the sum between a semibasic k–form and a nonsemibasic k–form, and the decomposition is unique. Two important properties are



(3.2.20)

for

and



The form

is degenerate and its nucleus is formed from vertical vectors. The equation

has a

solution if and only if

is semibasic, according to lemma 3.2.10. The

solution is not unique, it is determined by disregarding the addition of a vertical ield. The

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♭

♭:

isomorphism in Statement 3.2.5, restricted to

, de ines an isomorphism

Indeed, if

ield can be associated to it so that

the horizontal (unique)

Reciprocally, if

than the

1-form is determined through

and it is semibasic.

Using the previous isomorphism, we will now provide an intrinsic de inition of the Poisson Bracket and we will study some of its important properties.

De inition 3.2.16 Given f and g



(3.2.21)

where

. Poisson Bracket is the function de ined by:

and

are the (horizontal) Hamiltonian ields associated to functions f and g.

By using (3.2.20) one obtains: expression for Poisson Bracket is expressed by

and therefore a new

(3.2.22)



If ( expressions:

) are the arbitrary local coordinates on M, then we have the following local

(3.2.23) particularly, if



are Darboux coordinates on M then



(3.2.24)





Statement 3.2.17 The set can be structured as a Lie algebra. Proof. Indeed, the application { , } is bilinear and antisymmetric. To prove Jacobi’s identity, we

will irst calculate the value of the ield

By using the identity:

(3.2.25)



the conditions (3.2.6) that de ine the ield get:

and

as well as the relationship

(3.2.26) Therefore,



(3.2.27) and taking into account that:



(3.2.28) it results:



(3.2.29) relationship that implies:



(3.2.30)



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we

ield is horizontal. Jacobi’s identity is then a consequence of the fact that

closed:

is

and of the relationship (3.2.30).

Observation

The relationship (3.2.30) shows that the application

antiomomorphism of Lie algebras. The set

where

is an

is the relationship de ined by

equipped with the Poisson bracket { , }, is a Lie algebra that can be identi ied with a subalgebra of the Lie algebra through the (anti)isomorphism Further we will deduce the geometrical characterization of the Poisson Bracket theorem, in agreement with which the evolution in time of a system is generated by a Hamiltonian if and only if, for any pair of dynamic variables R, S the following relationship occurs (3.2.31) which in geometrical terms is written



(3.2.32) where X is the ield that describes the dynamics. As you have seen in theorem 3.1.49, for systems independent of time, the relationship (3.2.32) is equivalent to the property of ield X of being a Hamiltonian local. The next theorem presents the extension of this equivalence in the more general case of the nonautonomous systems.

Theorem 3.2.19 Given



(3.2.33)

a ield of vectors. The relationship

occurs for any pair of functions

if and only if the Y ield preserves the horizontal

distribution (meaning ields.

and



Proof. Given

and



(3.2.34) Also, the following equalities are obtained:



(3.2.35)





expression is found for



are cancelled for any pair of horizontal

the horizontal ields associated to functions f and g. We have:

in which we have taken into consideration that

and

A similar

and replacing in (3.2.33) we have the relationship:

(3.2.36)



Let us now assume that dt(Y) depends only on t and that is cancelled for any couple of horizontal ields; the relationship (3.2.33) is then true. Reciprocally, if (3.2.33) occurs, necessarily and coordinates’ functions.

for any couple of functions



, therefore also for

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f

f

because the

Observation 3.2.20 Let us now observe that the irst of the two conditions from the statement,

meaning

means that the Y preserves Kerdt, or equivalent, that the horizontal

distribution

is constant through Y, meaning

Related to the second

relationship, it can be noticed that for those ields Y that invary (which actually means symplectic variety.

Given now

cancelling the

on

is the analog of the condition of being local Hamiltonians in a an arbitrary ield that is written in canonical coordinates

(3.2.37) then the irst condition in the theorem, which characterizes only the vertical part of Y, tell us that c must be dependent only on t, c = c(t). The second condition characterizes only the horizontal part of Y and reverts to (3.2.38) Observation 3.2.21 By using the relationships (3.2.24) and the equations (3.2.12) it is deduced that

(3.2.39)



for From this it results that a necessary and suf icient condition that a function f to be a prime integral of motion (a preservation law) is that: (3.2.40) Under the conditions of theorem 3.2.19, it is noticed that if f and g are two prime integrals of the motion described by the Y ield then their Poisson bracket is also a prime integral (Poisson’s theorem). From theorem 3.2.19 is it understood that those Y ields for which are admissible for describing a time dependent Hamiltonian dynamic. In this case, their integral curves are parametrized by means of an s parameter that does not correspond to the t time, however along the integral curves. The next statement studies the particular case of the ields for which and gives a more direct characterization of the local Hamiltonian behavior for them.

Statement 3.2.22 Given

satisfying the condition

Lagrange 2-form associated to ield Y:

Proof. If L

where

the



is cancelled for any pair of horizontal ields, then mandatorily However, because

ield is a characteristic ield for

we have:

= 0. The Y

and therefore

and even though

Reciprocally, if

then

is cancelled for any pair of horizontal ields.

De inition 3.2.23 A diffeomorphism application if it satis ies the condition:

and



Then the property is equivalent with L

which preserves time

(3.2.41)

is called Poisson



For any pair of functions Observation 3.2.24 Poisson Bracket induces a Poisson structure on the V variety. Indeed, the (Leibniz) property:

f

f

f

f

f

f

f

f

f

f

f

f

f

150

f

f



(3.2.42) results from (3.2.22). The antisymmetric tensor twice counter variant structure) is given by (3.2.43) its rank is 2n and

is generated by dt. A

(which de ined the Poisson

Poisson application is de ined then equivalent through



, meaning invaries the Poisson structure An important characterization of the Poisson transformations is expressed in:



Theorem 3.2.25 A diffeomorphism

and only if there is a semibasic 1-form (3.2.44) Proof. The condition for the Poisson Bracket:

which preserves time is a Poisson application if so that:

to be a Poisson application may be written using the de inition of

(3.2.45) or equivalent:



(3.2.46) The left side of this last relationship may be rewritten so that the relationship becomes:

(3.2.47) and this shows that

is a Poisson application if and only if:



(3.2.48)



Because



(3.2.49)



Let us now assume that there is a semibasic k 1-form so that

with

by applying the

we obtain:

it results

by replacing

operator in the equality

Considering that

By contraction

is horizontal (because

preserves time),

in the relationship (3.5.49) it becomes:

(3.2.50) By replacing in this last relationship the expression found for the k 1-form one obtains:

(3.2.51)



Because is horizontal, from the last relationship it results therefore is a Poisson application. Reciprocally, let us assume that the relationship:

Considering that

(3.2.52)

into:

(3.2.53)



f

151

f

is transformed through

,

f



şi că

and

we obtain the property that

is cancelled on any pair of horizontal ields

there is a semibasic k 1-form so that

(or equivalent

Statement 3.2.26 Given part of a closed 1-form.

; then

Proof. Given

therefore,

is closed if and only if

is locally the semibasic

the local expression in canonical coordinates of the form

then

is closed if and only if : (3.2.54) which means there is an H function de ined on an open from V so that

(3.2.55)





This means that locally The converse is obvious. Observation 3.2.27 Let us notice that the semibasic k 1-form in theorem 3.5.25 veri ies the

condition

and thus according to the previous sentence, there is locally a

function so

that Additionally, the characterization given in theorem 3.5.25 for the Poisson applications, partially coincides with one of the de initions given to the canonical transformation [1] (some authors additionally require that the k 1-form to be exact).

Statement 3.2.28 Given

a ield of vectors so that

Then

if and only if

Proof. Indeed, if



we have

The left member of the last equality of

a semibasic 1-form, therefore mandatorily Statement 3.2.29 Given pair of horizontal ields.

and

. ; then

is closed if and only if

cancels itself for any

Proof. The statement is easily proven if we use statement 3.5.26. Indeed, if

is mandatory that

where

is a semibasic 1-form and thus

Reciprocally, is then there is

so that

cancels itself for any pair of horizontal ields,

and therefore

Observation 3.2.30 Let us assume that pair

then it



satis ies the relationship

From statement 3.2.28, considering that

for any

and that

it results that

From statement 3.2.26 it results that locally there is a function H (Hamilton’s) so that and therefore it is obtained that 1, then Z is a characteristic ield for

If additionally we assume that dt(Z) =

and in a system of canonical coordinates



(3.2.56)



Let us also remark that the structure

it is written



meaning V is of odd dimension and

is a contact variety in the sense given in [1],

is a presymplectic form of maximum rank. If

application then

and we can associate to the contact structure

contract structure

where

şi

for which

f

f

f

f

f

f

f

f

f

152

is a Poisson a new

Reciprocally, if

preserves time and satis ies the relationship

for which dt(Z) = 1, then

for any

is a Poisson application.

Observation 3.2.31 From statement 3.2.22 and observation (3.2.30), it results that if satis ies the condition dt(Z) = 1, then the following properties are equivalent:



(3.2.57)



3.2.2 Forms with variations associated to the irst order differential equation systems A way to introduce the forms with variations, in the case of irst order differential equation systems, is that by which the presentation of the theory of formulas with variations is reproduced, made in § 2.2, in the case of systems of this form. Another way is the one we will follow, further along, by which the irst order equation systems are regarded as a particular case of second order systems, in which the second order derivates are missing. 1 Geometrical interpretation Let us take as given, together with the dynamic system (4.2.64), the Cij, Di coef icients. At a local map change, these coef icients change according to the formulas: , , and therefore, constitute the components of an antisymmetric covariant twice distinguished tensor and those of a convector, respectively. Let us now form, with these coef icients, 2-form: . The property of the v 2-form of being closed (dv =0) is equivalent with the conditions of autoadjunction (4.2.66). Observations 3.2.32 a) Any second order differential system of the form equivalent to the irst order system, canonically assisted:

is



b) In the case of systems written in fundamental form:

,

the irst order system is obtained:

, dij This is equivalent, in the det(Aij) ¹ 0 hypothesis, with the system in the form of:



:

from where:

, , . Making the notations: Cij = 0, Ci,n+j = Aij, Cn+j, i =-Aij, Cn+i,n+j = 0 it is deduced that the equations (4.2.66) are equivalent to:

Aij =Aji , relationship based on which it is obtained:



f

f

f

f

f

f

f

f

f

f

153

Statement 3.2.32 Given canonically associated second order system:

,

a second order system and the

, , If the last one is auto-adjunct, then the given system is auto-adjunct. 2 Hamilton’s equations’ auto-adjunction Given Hamilton’s function, de ined by: H = piyi(t,q,p) – L[t,q,y(t,q,p)], formula from which Lagrange’s function L is explained: (3.2.67) L = piyi(t,q,j(t,q,v)) – H[t,q,j(t,q,v)]=ji(t,q,v)qi –H[t,q,j(t,q,v)], Let’s now consider Lagrange’s function (4.2.67), let us put the integral’s extreme condition: A c(e) = , and replace the L function with its expression (4.2.67). From the extreme condition it is obtained:

/e=0 =



-



+





Considering the conditions imposed at the ends of the integration interval:

as well as that the above considered integral must be null, no matter which are the variations



şi

, Hamilton’s canonical equations are obtained:

(3.2.68) , . Statement 3.2.33 The equations (4.2.68) are autoadjunct. The conditions are veri ied (3.2.66). 3.2.3 The direct problem and the reverse one in nonautonomous Hamiltonian dynamics

In this subparagraph we will handle the direct problem and the reverse one [xx] for nonautonomous Hamiltonian systems. We will use the notations and conventions made in the previous paragraph.



1 De initions and examples Given (

) a nonautonomous dynamic system, and

vector ield X dependent on time.

is compatible with the cosymplectic structure

If Y satis ies the equivalent properties (3.3.67).

The direct problem returns when giving a symplectic structure

associated cosymplectic structure

and of a

f

f

f

f

f

together with the

1-form satisfying the equivalent

154

f

the extension of the



De inition 3.2.34 We will say that (where



a symplectic variety, .

properties from statement 3.3.29 and the determination of a horizontal ield The ield Y, extension of

so that

is then compatible with the structure

and

Hamilton’s function is de ined according to sentence 3.3.26 through The direct problem has a unique solution and its solution is a consequence of the existence of the isomorphism ♭: the restriction of the ♭ isomorphism from statement 3.3.6. The reverse problem in time dependent Hamiltonian dynamics consists in giving a nonautonomous dynamics system on an M variety of size 2n and determining a symplectic 2-form

so that ield Y, extension of X, is compatible with the associated

cosymplectic structure The reverse problem does not always admit a solution, as we will see in the examples given at the end of this subparagraph. Further, we will approach locally the solution of the reverse problem, in a similar way to that in paragraph 3.1.

Given the M variety of size 2n and

a system of local arbitrary coordinates on M. The

time dependent vector ield X admits the local writing nonautonomous equation system:

(3.2.69)



It is known that an integrating factor

and de ines the

can be determined so that the equation system:

(3.2.70) equivalent to the irst, is autoadjunct. The autoadjunction conditions are the general ones (3.2.66): where we denoted

and are equivalent to the 2-form property of being closed. Let us assume that we have found a time

independent nondegenerate solution

of the system (3.2.70); then the 2-form

(3.2.71)



de ines a (local) symplectic structure (local) on M. Additionally, if

is X’s extension, then

is actually the Lagrange 2-form associated to the Y ield: the condition

where

;

being satis ied, it results that Y is compatible with the cosymplectic structure

. Therefore, the local solution to the reverse problem reverts to determining a time independent integrant factor. Observation 3.2.35 The system (3.2.66), to which we add the condition to the system (more comfortable to use in practice):



(3.2.72)



If dimM = 2, denoting with

is equivalent

the system (3.2.72) is reduced to the equation:

f

f

f

f

f

f

f

f

f

f

155



(3.2.73)



It is noticed that any horizontal ield under the form admits a time independent integrant factor and, therefore, admits a time dependent Hamiltonian representation. Example 3.2.36 Given the dynamic system (M=R2):



A time independent integrant factor is

from R 2 ) is

The 2–symplectic form (de ined on an open

and the Lagrange 2-form associated to the

ield

is: Correspondingly, one obtains Hamilton’s function Example 3.2.37 Given the dynamic system:





The X ield is in this case Hamiltonian canonical (divX = 0

) and as such:

and Example 3.2.38 Given now the equation system: where we should assume that the functions

and

satisfy, on a real interval I, the condition

Let us assume that the system could admit a time independent integrant factor on a U open from R2. It is then necessarily obtained (from (3.2.73)):



(3.2.74) Deriving partially related to t the equality (3.2.74) it is obtained:

for any and which is absurd. As such, in this case the reverse problem does not have a solution. Example 3.2.39 Let us now consider the equation (the harmonic oscillator with amortization):



The equation is equivalent to the system:



f

f

f

f

156

where we have denoted with condition

We will show that if the functions c(t) and k(t) satisfy the

on a real interval I, the reverse problem does not have a solution. Indeed, if

the system admitted a time independent integrant factor

on a U open from R2, then from (3.2.73)

it is necessarily obtained: and conveniently arranging, one reaches the equality:

from where, partially deriving related to t



Deriving once again partially related to t the last relationship we ind

matter what might be

şi

no

which is absurd.



3.2.5 Some notions about the Hamiltonian formalism



1 Retranscribing Hamilton’s equations under geometric form: Using the denotations:

, where qm are the Cartesian coordinates in which the Newton’s equations are written. Newton’s equations can be written under the form:

(3.2.75) The functions

m

are considered as being the components of a ield of vectors on T(T*M)):

m

(dependent of the t parameter). The formulas (3.2.75) represent the counter variant form of Hamilton’s equations. Their covariant form is written as follows: where are the components of the inverted matrix of 2 Observations a) The analytical character of Hamilton’s equations is expressed through their property of resulting from Hamilton’s variational principle on phases space. Let us consider the action: A=



which is also written, using the notation:

The variation of the action is:

A=



dA =

=

=



f

157

f



under the form:

=





From the property dA = 0, "dt, "dam, it results:



from where: . To conclude, from the variational principle, the covariant form of Hamilton’s equations results. b) The algebra character of Hamilton’s equations The counter variant tensor wab is called: the fundamental cosymplectic tensor of space. With its help Poisson’s brackets are de ined and built, like this: Given the function A:aÎT*M®A(a) ÎR, whose derivative along a curve on the phases’ space is written (a) = Considering the counter variant form of these, we can write: (a) =

[A,H].

With the help of Poisson’s brackets, Hamilton’s equations become: Generally, Poisson’s bracket for the functions A and B is de ined by:

=[

,H].

[A,B] = . Poisson’s brackets check the relationships: [k1A1 + k2A2, B] = k1[A1 , B] + k2[A2, B], [A, B] + [B, A] = 0, [[A, B], C]+[[B, C], A]+[[C, A], B] = 0. c) The geometrical character of Hamilton’s equations If one considers the covariant tensor wmn (fundamental symplectic tensor), it de ines the canonical symplectic structure of the T*M space, through the 2-form:

w = wmndamÙdan = dpkÙdqk. If it is considered the canonical 1-form (Liouville): q=

and the action:

dan = pkdqk,

A= where



is called the integral invariant of Poincaré-Cartan, can be written:

, Assuming the autonomous system H = H(a), Hamilton’s equations are written globally under the form: iXw = -dH.

§ 3.3 NONHAMILTONIAN DYNAMIC SYSTEMS As is has been mentioned, a pivotal role in the study of dynamic systems is held by the discovery of new preservation laws (prime integrals) expressed by means of functions whose

f

f

f

158

3.3.1 Almost cosymplectic and bicosymplectic structures associated to a nonhamiltonian dynamic system 1 Almost cosymplectic structures

We will show below how special geometrical structures can be associated to a nonautonomous

vector ield where V = R ´ M and M is arbitrary differentiable variety of 2n size; related to these the ield X has certain geometrical properties that allow for the tracing of preservation laws associated to the movement described by X. In [7], [9] there were presented methods for tracing of preservation laws for autonomous bihamiltonian dynamic systems. We will generalize (see [26]) these methods for nonautonomous dynamic systems, in the most general form. De inition 3.3.1 Given M and arbitrary variety of 2n size and V = R´ M the product variety. It is called almost cosymplectic structure a couple semibasic 1-form (meaning

where

and

is a volume for on V.

Observation 3.3.2 As the almost cosymplectic structure

almost contact structure, there is a unique ield

is at the same time an

(Reeb’s ield) that satis ies the conditions



(3.3.1)





Obviously in this case



Statement 3.3.3 Given an almost cosymplectic structure

and

ield dependent on a parameter (regarded as a horizontal ield on V, unique ield

(3.3.2)



Proof Indeed, given

From the irst condition it results that A = 1, and from the belongs to the nucleus of the volume 2–form



(3.3.3)





De inition 3.3.4 A vector ield

for which dt(Z) = 1 is called compatible related to the If the Lagrange 2-form

is closed:

f

f

f

f

f

f

f

f

159

f

and

Consequently, the extension of the X ield is

almost cosymplectic structure

f

There is then a



therefore

f

a vector

called X’s extension, that satis ies the properties:

second one it results that

f

is a

and semibasically closed (

having the property that

f

f

derivatives along the trajectories are null. Several methods for determining preservation laws are known. We quote in the Lagrangian formalism the method supplied by the Noether theorem [25]. In the Hamiltonian formalism can be obtained, as we have seen, preservation laws by using the canonoid transformation on symplectic varieties (in the autonomous case) or cosymplectic (in the nonautonomous case), introduced by Saletan and Cromer in [50]. In [7-10] Cariñena and Rañada extended the determination method for such laws for bihamiltonian dynamic systems presumed to be generally autonomous. Further, the procedures for determining prime integrals in nonhamiltonian dynamic systems are expanded. The reverse problem in the formalism of almost symplectic structures is raised and solved and the adoption of the Cariñena şi Rañada method is proposed also for nonhamiltonian dynamic systems of very general form. It is shown that, locally, such laws can be traced for any dynamic (differentiable) system.





Observation 3.3.5 In the case when

obviously that about X.

and if Y is compatible with the structure

Given vector ield

there is an extension of a horizontal ield X, we will say the same

a local arbitrary coordinates system on M in which the nonautonomous X admits the local writing X=

dependent) can always be determined

An integrating factor (generally time so that the equation system:

(3.3.4) equivalent with the nonautonomous system



(3.3.5) is autoadjunct. The autoadjunction conditions are in the most general case of the form (3.2.66), where we denoted with

Let us now de ine the 2–form:

(3.3.6)



and take = the Lagrange 2-form associated to the Y expression of the X ield. Because the conditions (3.2.66) are equivalent to the property of this form of being closed and considering the fact that to any system of the form (3.3.5) can be associated an autoadjunct equivalent system (and therefore Lagrangian) of the form (3.3.4), we can state: Statement 3.3.6 For any time dependent vector ield X there is (locally) an almost cosymplectic structure related to which X is compatible and a cosymplectic one whose Reeb ield is X’s extension. Observation 3.3.7 The previous statement solves (locally) the reverse problem for nonautonomous dynamic systems. Practically, one asks the question of determining nondegenerate solutions of the system (3.2.66) or of the equivalent system.



(3.3.7) If dimM = 2, then the system (3.3.7) is reduced to the equation:

(3.3.8) There is also another way by which we can associate to a nonautonomous dynamic systems such a special geometrical structure: Example 3.3.8 Let us consider the differential equation: (3.3.9) equivalent to the system:





(3.3.10)

where We will denote with:

and the functions c(t) and k(t) satisfy the condition imposed in example 3.3.3.

f

f

f

f

f

f

160

(3.3.11) the local representation of a diffeomorphism to a system of the form:

that preserves time. With its help we are led

(3.3.12) for which one can obtain the geometry of the system as well as the evolution constants as images of the system’s corresponding elements (3.3.10), and reciprocally. Let us seek out a diffeomorphism through which the system (3.3.10) to be taken into the system (example 3.3.4):

(3.3.13)





The differential equation associated to the system (3.3.13) is

(3.3.14) and coincides with the equation (3.3.9) if:



:



(3.3.15



This occurs if and only if know (example 3.3.4) the cosymplectic structure de ined by:

For the system (3.3.13) however we



(3.3.16) Coming now back to the system (3.3.10) through the automorphism :



(3.3.17)



we obtain the cosymplectic structure associated to the ield given in statement 3.3.6, de ined by:

(3.3.18) Consequently, the

(kx1+cx2)

in the meaning

ield X is compatible related to the almost cosymplectic structure

We will now prove a result useful in applications, similar to the one in observation 3.3.24; this time, however, we no longer dispose of a cosymplectic structure on an M base variety.

Statement 3.3.9 A vector ield

cosymplectic structure

for which dt(Z) = 1, is compatible with the almost

if and only if

(3.3.19)



Proof. Because şi Z is Reeb’s ield of the cosymplectic structure it results from De inition 3.3.4 that (3.3.19) is true. Reciprocally, let us assume that (3.3.19) occurs. A direct calculation proves that

the semibasic

(3.3.20)



f

f

f

f

f

f

161

f

form being given by:

where

It results from (3.3.19) that

we obtain

and therefore

from where,

being semibasic,



Observation 3.3.10 (The structure of compatible ields) Let us assume that ield (the extension of a horizontal ield X) is compatible related to the almost cosymplectic structure There is then (locally) a potential of the Lagrange 2-form

(3.3.21)



the differentiable functions



(3.3.22)



Given

and

are solutions of the system

a ( ixed) solution for the irst group of equations (3.3.22), G(t,x)

arbitrary function and

an

a vector ield so that:



(3.3.23)



The Z is compatible with the almost cosymplectic structure

it is noticeable that

because, calculating

where



(3.3.24)



Reciprocally, for any vector ield Z compatible with the structure



veri ies (3.3.23), where derived equation system:





there is a function G that

is the solution previously set. Indeed, we are naturally led to the partially

(3.3.25)



for which the compatibility conditions

are satis ied because of the way in which

the functions were chosen and because are solutions for the third group of equations from (3.3.24). Obviously, G is determined by not considering the addition of a g(t) differentiable function. 2 Canonoid transformations Let us now generalize the notion of canonoid transformation and extend the results in §3.2 for when we do not have a symplectic 2-form on the base M variety.

De inition 3.3.27 Given

structure

a vector ield compatible related to the almost cosymplectic

A diffeomorphism

that preserves time is called canonoid

transformation related to Y if the ield

is compatible related to the same structure



It results that the property dw+d(iXw)Ùdt = 0 implies the property dw+d(i



By de inition, it results that the forms d(iXw) şi d(i

f

f

f

f

f

f

 

f

f

f

f

f

162

f

w)Ùdt=0.

w) have equal semibasic (horizontal) parts,

meaning, iXw şi i w differ by a semibasic differential. 3 Dynamic systems with two state parameters

f





Let us irst assume that dimM = 2 and given the horizontal ield X and Y its extension. In this case

where the integrant factor canonoid transformation represented locally by:

veri ies the equation 3.3.6. Let us assume the

(3.3.26)



we then have:

which shows that

where

şi is a semibasic 1-form. Therefore, the relationship (3.3.6) remains true, as well as the theorem 3.3.3; the demonstrations are transposed without dif iculty. If dimM > 2, the method used in § 3.2 to determine the preservation laws is immediately adapted and the statements 3.3.5 and 3.3.6 are demonstrated analogously.

Moreover, because to ield Y it is associated the cosymplectic structure

de ine the Poisson bracket of two functions

by

(3.3.28)

where



are the Hamiltonian ields associated to functions

uniquely de ined by



(3.3.29) In local (arbitrary) coordinates we have the expression



(3.3.30)



If dimM = 2, the preservation law will have the expression



equivalent, coef icient

we can

= constant, or

constant. In the case when dimM > 2, the preservation law associated to the from the relationship (3.3.10), adapted in this case, is

where

Canonoid transformations generating functions We will now show that a canonoid transformation can be determined by giving a generating

function, extending thus the results in §3.2 and §3.3. Let us assume that

is a canonoid

transformation related to the ield Y given by (3.3.3) and take the vector ield satisfying the property (3.3.19). Considering the ideas expressed in Observation 3.3.10, we can (locally) write the relationship

(3.3.32)



where is given by (3.3.25) and F is a differentiable function on an open from V called the generating function of the transformation Reciprocally, let us assume a Y ield compatible related to the almost cosymplectic structure system (3.3.23). Then we have:

and take

Statement 3.3.30 If differentiable functions so that

a ixed solution of the irst group of equations of the

is a diffeomorphism that preserves time and

f

f

f

f

f

f

f

f

f

f

f

f

f

f

163

F, G are two

(3.3.33) then is a canonoid transformation related to Y. Proof. By differentiating both members of the relationship (3.3.33) and using (3.1.30) one obtains: (3.3.34) from where it results that locally there is a differentiable function W so that

(3.3.35)





From (3.3.35) we obtain (denoting with



(3.3.36)

):

therefore and is a canonoid transformation. Observation 3.3.31 The relationship (3.3.33) is equivalent to the equation system with partial derivatives:

(3.3.37)



where we denoted with If the generating function F is given then the transformation is determined by the irst group with 2n equations from (3.3.37) and the function G by the second equation. Let us assume that we have a solution

that veri ies the irst group of

equations in (3.3.36). Then the ield is uniquely determined and we can consider the equation (3.3.26) in the context in Observation 3.3.10. As it will result from example (3.3.14), a G function determined from the last equation (3.3.37) is not always a solution for (3.3.26). However the function G + W veri ies (3.3.26), W being de ined by (3.3.35). Observation 3.3.32 The previous results generalize the ideas in §3.2, if a symplectic structure can be determined on the space of M phases. Indeed, by using the expression: with

(3.3.38)



and B as in observation 3.3.10, we obtain: (3.3.39)



If we assume that the integrant factor does not depend on time and also then keeping in mine (3.3.38), the irst group of 2n equations in (3.3.37) can be written in a similar form with (3.3.31), function B playing the role of Hamilton’s function H. Example 3.3.33 Let us now revert to 3.1.9 keeping the hypotheses and notations made § 3.1. Given thus the equation: (3.3.40) and the equivalent equation system:



(3.3.40') An almost cosymplectic (global) structure associated to ield X is given by

(3.3.41) and the corresponding cosymplectic structure is:



f

f

f

f

f

f

f

f

164



(3.3.42)





A potential for the Lagrange



(3.3.43) For F = 0 in (3.3.36) the canonoid transformation is obtained related to X given by:

2-form is

(3.3.44) and the associated preservation law is:

(3.3.45)



The ield



(3.3.46)

constant.

the Lagrange

2-form and its

potential are respectively:



Therefore, the triple that veri ies the relationship (3.3.32) in Statement 3.3.12 is composed of the canonoid transformation given by (3.3.44) and the functions G = F = 0. The function G = 0 also veri ies the system (3.3.26) where last equation in (3.3.36) becomes

are given by (3.3.43) and

(3.3.47) and admits for example the solutions

by (3.3.46). the



(3.3.48)



The functions do not verify the system (3.3.26) and the difference is not a function that depends only on time; these facts are in agreement with the statements in observation 3.3.13. Observation 3.3.34 If instead of the relationship (3.3.32) we choose as de inition of the generating function the relationship:

(3.3.49)



then F represents the deformation of the Lagrange function transformation. Indeed, the function the dynamic system (3.3.4) given, and system de ined by

generated by the canonoid

is the Lagrange function calculated on the trajectories of is the Lagrange function associated to the dynamic



3.3.2 Dynamic systems on bicosymplectic varieties [44].

We will now generalize the expressed method for determining the preservation laws without appealing to the use of canonoid transformations. Let us assume as given the almost cosymplectic structure and the vector ield Y (through (3.3.3)) compatible in relation to it. 1 The direct problem of dynamics in bicosymplectic formalism

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Given a differentiable variety M of even dimension 2n and with it the variety V=R´M. Given on the

V variety a almost cosymplectic structure (

), so that hn=wÙq is a volume form. We will assume

that the w form is semibasic and semibasic closed:

. A (nonautonomous) dynamic

system on M is a ield X:R´M®TM. To this it is associated Reeb’s ield Y = X + ÎX(V) and Lagrange’s form: WY = w+(iXw)Ùdt. We will say that the Y ield is compatible with the almost cosymplectic structure (q,w) If dWY = 0; from where it results that Thus, given the dynamic system (M,X) on the variety V=R´M, a cosymplectic structure was introduced (q, WY). De inition 3.3.35 A if it is invariant through Y, meaning:

2-form is called admissible for the dynamic system

(3.3.50) We will call a bicosymplectic weak structure associated to the given Y ield the ensemble

in which the admissible form is degenerate, and bicosymplectic if is nondegenerate. It is called a local dynamic system and bihamiltonian weak on the bicosymplectic weak variety (M,q,w,W,W') a ield X:R´M®TM, for which Reeb’s ield Y = X +

veri ies the relationship WY =W,

meaning: W=w+(iXw)Ùdt, where w=W- WÙdt), and for which the W' form is admissible. The dynamic system X is called bihamiltonian local if the variety (M,q,W,W') is bicosymplectic. If (M,q,W,W',X) is a local dynamic and Hamiltonian weak system, then all the 2-forms de ined by W - lW', lÎR, are admissible for the given system, so that the forms hl=(W-lW')nÙq (except at most a inite number of values of l) are volume forms and therefore: The function f:R´M®R bears the name of characteristic function of the local and bicosymplectic



weak structure (M,q,W,W'). The coef icients of the function are preservation laws associated to the given dynamic system (the proof is similar to the one in §3.2).

Observation 3.3.36 The role of the Lagrange 2-form

form

can be taken by a nondegenerate 2-

admissible in relation to Y. Consequently, given the X ield, the (local) solution to the

reverse problem reverts in this case to determining two integrant factors

(nondegenerate) and

for the system (3.3.7). 2 The reverse problem in bicosymplectic formalism Given a differentiable variety M of size m=2n and on it a vector ield dependent on a parameter and X:R´M®TM. Let us determine R´M a bicosymplectic structure (M,q = dt,W,W') for which the extension Y of ield X to be compatible with the cosymplectic structure (q,W) and W' to be admissible for (M,W,q,X). Theorem 3.3.37 The reverse problem always admits (locally) a solution. Proof To ield X it is associated (locally) the differential system which admits integrant factors of the form Cij(t,x), with which it transforms into an equivalent and autoadjunct system: + Di =0. The autoadjunction conditions are given by: (3.2.66). All the solutions of this system, with a non-null determinant, constitute an integrating factor. Denoting through

, Lagrange’s form is written: W = w + iXw Ù dt,

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f



whose characteristics are the trajectories of the given system. Given Cij and C'ij two solutions of the system (3.2.66), we respectively have the forms W and W' and therefore the bicosymplectic structure (M,q,W,W') for which ield X is compatible. Building the volume form (w-lw')n Ùdt = flwnÙdt, these prime integrals are obtains: f1,...,fn, so that fl = 1+f1l+...+fnln. 3 The case of the systems with two state parameters Given a system with two state parameters:

let us look for an integrating factor under the form: .

Let is rewrite the equivalent autoadjunct system:



The only equation that this integrating factor must satisfy is:

Let us assume as known, for this equation, two integrating factors: x1 and x2 non-null. These are associated with, respectively, the following two Lagrange forms: Wi = xi[dx1Ùdx2 + (f1dx2 – f2dx1)Ùdt], that supply, with q = dt, a bicosymplectic structure and the following two volume forms: hi = xidx1Ùdx2Ùdt, which are, obviously, proportional: h1 =Fh2. From this relationship is results that x1 = Fx2 and therefore F = . The F function obtained like this is a prime integral of the given system. Example 3.3.38 Given the decoupled system:



(52) The equation that gives the integrating factor is:





We have the adjunct system:



Firstly, looking for a solution independent of xi, it is:



A second solution independent of t is:





therefore, the preservation law is: (54)



constant.,

which is easily veri ied if the general solution of the system is considered: x1 =



§ 3.4 HAMILTONIAN FORMALISM OF THE CONTINUOUS ENVIRONMENTS The reverse problem of system dynamics with a inite number of degrees of freedom is completely solved and beautifully presented by Santilli in [51]. The extension of this problem to the

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case of the continuously deformable environments and especially to the theory of the ield, initiated by Santilli himself in [52], has been resumed by several authors, of which we quote: Helmholtz, Tonti, Anderson, Henneaux. This paragraph proposes, in general, to present methodically the analytical study of the dynamics of the continuously deformable environments, to present techniques for obtaining preservation laws associated with these types of dynamic systems and to study the inverse problem. The principles and methods of analytical mechanics used by us in previous chapters to study the evolution of dynamic systems with a inite number of state parameters can be extended to dynamic systems of continuously deformable environments. Bearing in mind, however, that these systems possess an in inite number of state parameters, consequently the differential equations describing the evolution cannot take the form of the Euler-Lagrange or Hamilton equations in the chapters dedicated to them, since their number, which coincides with the state parameters number, would be in inite. For this reason it is necessary to determine a system of Euler-Lagrange or Hamilton equations, other than those known and formulated in the quoted place, for continuously deformable environments. Such equations (with partial derivatives) will also be useful in studying ields such as electromagnetic ield, generalized relativity theory, and quantum mechanics. E. Noether showed that when evolutionary equations derive from a variational principle (called the Hamilton principle), a systematic procedure can be established to detect preservation laws directly from this principle. We will now present such a situation in the case of continuously deformable environments. In this paragraph the posed problems, the results obtained as well as the calculation methods copy, by generalization, their correspondents in the previous chapters . 3.4.1 Canonical equations for continuously deformable systems Hamiltonian formalism, in the mechanics of continuously deformable environments, is introduced by analogy with that of material point systems with a inite number of state parameters. In the case of the continuously deformable environments it is available a Lagrangian density L = L(

), generally, de ined on J1(R´N,M). Derived functions:

(3.4.1) ( ), bear the name of impulse density, associated respectively to the variational parameters ua. The formulas (3.4.1) can be named, by generalization, Legendre transformation. Now, Hamiltonian density can be de ined by: (3.4.2) H= L The number of terms of a sum in (3.4.2) is given by the number of ield variables that de ine the system and not by the number of variables. This relationship becomes a function as follows: Take c:I´D D

an application, then:

by:

(3.4.3) H = H[ ], (i,h= . Consequently, the Hamiltonian of an continuously deformable environments will be described



(3.4.4) H= Let us now consider a C arbitrary variation of c: C :J´I´D

D

and let us calculate the



.

, (e,t,x)®(t(e),x,



derivative under the condition that xi remain ixed, we will have:

(3.4.5)



To calculate the derivative in a different way, we will consider the H de inition and the expression of the Lagrangian density L, we will obtain:

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(3.4.6)



Based on the de inition relationships (3.4.1) of the impulse density, the irst and last term (3.4.6) are reduced. Let us then perform the integration by means of parts of the penultimate term:

, therefore:

The irst integral of the right member can be transformed, based on the theorem of Green-GaussOstrogradski, in a surface integral, and based on the fact that the variational parameters ua do not vary on the ¶D frontier, the integral is cancelled:



We, therefore have, for the derivative of an H variation, given by (3.4.6), the expression:

,

or, based on the Euler-Lagrange equations:



(3.4.7) . Calculating the H variation, given by (3.4.5), by using the integration by analog parts, we get to:

(3.4.8)



Equalizing the coef icients of the arbitrary variations (3.4.7) and (3.4.8), leads to the equation system:

(3.4.9) as well as to the identity

,

,

, from the expressions



(3.4.9') known also in the case of the systems with a inite number of state parameters. The equations (3.4.9) in 2m number are analog to the Hamilton’s canonical equations in the case of the discreet and inite environments systems. 3.4.2 The symmetrical form of canonical equations As it can be observed, canonical equations (3.4.9) do not bene it from the property of having a symmetrical form, analog to those in the case of the systems with a inite number of state parameters, this is why we will seek to change the way they are presented. With the help of operators called functional, variational or eulerian derivatives:

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, applied to an F ''density'', Euler-Lagrange equations, of the variational problems are written:

Keeping in mind that the H Hamiltonian density does not depend on

derivatives of H, related to

and

and

, the functional

, are respectively:

expressions with whose help the canonical equations receive the form:

(3.4.10) form which is now symmetrical; the equations (3.4.10) bear the name canonical equations and correspond to the classical canonical equations. 3.4.3 Poisson brackets for continuous deformable systems Another use of the functional derivative, besides the ones met until now, is to serve the de inition of Poisson’s brackets for the continuous deformable systems. Given the functional:

(3.4.11)

F=

whose ''density'' F is function of

and

. Let us assume the ixed area of integration D, and

the variables ua and satisfying the canonical equations (3.4.10). Let us totally derive in relationship to t the functional (3.4.11), we have:



.

Keeping in mind now the relationship:

= and using the theorem of Green-Gauss-Ostrogradski, based on which the prime integral of the right member becomes the surface integral:

which is cancelled by reason of the fact that ua does not vary on the ¶D frontier. With these results and based on the de inition of the functional derivative it is obtained:

from where, with the help of canonical equations:

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.

Thus, the expression:

(3.4.12) bears the name of Poisson’s bracket of the F and H functions, de ined for the continuous deformable systems. Thus: + . This relationship helps the study of ''prime integrals'' of the canonical system (3.4.10), by means of a procedure similar to the one met in the systems with a inite number of state parameters, as well as the theory of the ield in a special way. 3.4.4 Lamé’s equations in Hamiltonian formalism Let us now deduce Lamé equations in Hamiltonian formalism. For this, let us consider the Lagrangian density: L=

-

-

-

+

, (i,k=1,2,3)

where ui are variational parameters. For the impulse densities we have the values: with which the Hamiltonian density becomes:



,

H= -L = + + + This form of expressing the Hamiltonian density is not convenient, it is imposed to eliminate the

parameter

so that:

=

cu

, therefore:

H=

+

+

+

-

,

,

and the canonical equations lead to: Eliminating the functions to Lamé’s equations:

,



between these equations leads, through convenient terms groups, .

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Cap. IV HAMILTON – JACOBI FORMALISM The most general process of integrating canonical equations of evolution is the search for the canonical transformation that binds the conjugated canonical sizes q and p, considered at a certain point, by the initial values q0 and p0, meaning by a string of given 2m constants. In this way, the problem of the solution of the considered evolution problem is obtained under the form: (4.1.0) By studying the mathematical properties of such a contact transformation, Hamilton and Jacobi established a new method for solving the evolution problem of a dynamic system (particularly that of a particle), which proved to the especially ef icient in the case of conservative forces and of central ields. The theoretical importance of the method resides in the fact that it allows us to conceive classical mechanics from a totally new point of view, which proved to be fertile to the theoretical speculations regarding the analogy between classical mechanics and the geometrical optics. § 4.1 HAMILTON – JACOBI EQUATIONS The canonical transformation (4.1.0) is determined by the fact that the new Hamiltonian H must be identically new, therefore the new canonical equations will be written:

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Therefore we will have: and being two sets of arbitrary constants. From the de inition of canonical transformations it results that: (4.1.1) formula in which S is the generating function of the canonical transformation. Let us choose the generating function Hamilton. Then:

, called the main function of

, so that the following equation is obtained:

(4.1.2) known under the name of Hamilton – Jacobi equation. The equation (4.1.2) is an equation with irst order partial derivatives with m+1 independent variables: (t,qi). A general solution of it will have n+1 integration constants: αi.... αn+1, Given that in the equation (4.1.2) S’s function only appears by means of its partial derivatives, it results that if it is a solution of (4.1,2), then S+ is also a solution, in which α is an additive arbitrary constant. We notice the S function in (4.1.2), only appears by means of its partial derivatives, it results that if it is a solution of (5.1.2), then is also a solution, in which is an additive arbitrary constant. A complete solution of the equation (4.4.1) can be written under the form: in which no αi constant is additive. By deriving the integral S related to αi and equalising these partial derived integrals with the arbitrary constants βi a system of algebraic equations is obtained, whose solution allows us to discover the expression of the qi coordinates as functions of t and the coordinates αi and βi . The canonically conjugated impulses will be written under the form: (4.1.3) . In conclusion, Hamilton’s main function is to generate contact transformations, which supply us the 2 initial conditions αi and βi and allow for the writing of the evolution law under the form: . It can therefore be stated that the problem of determining the evolution of a system can be formally reduced to the integration of the Hamilton-Jacobi equation. This equation is equivalent to the canonical equations (Hamilton’s) The problem of determining the evolution of a dynamic system is formally reduced to the integration of the Hamilton – Jacobi equation, equation which is equivalent to the canonical equations, Hamoton’s. An important physical interpretation of the main function of S results from its total derivation:

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relationship that, together with (4.1.3) leads to :

,



so that

Therefore, the S function differs from the action of the system only by means of an additive constant. Thanks to this interpretation and with the help of the Legendre transformation, it was highlighted, by the theory of Hamilton-Jacobi, that time independent holonomies can be expressed with the help of the main function S, which represents a physical preservative size, with connections to the evolution of dynamic systems When the system’s Hamiltonian does not depend explicitly on time, then HamiltonJacobi’s equation becomes: by integrating the equation (4.1.2) related to t, it is obtained: , formula in which the function solution to the equation:

is called Hamilton’s characteristic function. It is a

and where E is the total energy of the considered system. From here, it also results the interpretation of the characteristic function. The W function is the generator of a canonical transformation by means of which the generalized coordinates move to cyclic coordinates .



§ 4.2 ACTION AS TIME FUNCTION



As it is known, a Lagrangian dynamic system is de ined by a function

which along any curve

, to

with the property c(t0)=x0, c(t1)=x1, points ixed in M, it is

associated the real number . The principle of minimal action states that the evolution of the system xi=xi(t) is done along that curve from , for which Ac takes an extreme value. This condition requires that the curve c is a solution to the Euler-Lagrange equation system: Another aspect of this problem tells us that the action is a size that characterizes the motion along the real trajectories with common origin.

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Let us consider various trajectories with the x0 origin with different inal positions. It results that the whole action on the real trajectories can be considered as a function by the coordinates of the points at the superior limit. If C is considered as a variation of theirs, we have:

(4.2.1) formula in which the bracket under the integral symbol represents the Euler-Lagrange equations that cancel themselves on the real trajectories. By multiplying part to part the relationship (4.2.1) with and considering that

we are led to: .



We will use the notations

and

and we have:



From these relationships it is deduced than on trajectories . Considering the ixed points x(t0)=x0 and x(t1)=x1, the action, calculated on various curves that tie the two points between them, has in the ixed point x1=x(t1), a value dependent of the t parameter that characterizes the curve; it is therefore a function of t: A=A(t). It can therefore be considered the determination of the function

.

From the de inition of the action function: the relationship:

it is deduced that



it results that:

(4.2.2) Consequently, it can be written:

.



as well as:



, and from







.

The minimum condition is that The equation:



, from where the (4.2.2) relationship is obtained.



,

where is the Hamilton –Jacobi equation. Considering a system of ordinary differential equations, of irst order, written for example under the cinematic form . There is an integrating Aij factor by multiplication with which the system transforms into one that is equivalent to it, auto adjunct

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To it a Lagrange is associated ,

for which its Euler – Lagrange equations are

,

. The Lagrange function

has the property that the action function

veri ies the relationship (4.2.3)

but

(4.2.4) From the identi ication of (4.2.3) with (4.2.4) it results

Therefore , or

and therefore, the action is



To the given system it is associated the equation (4.2.4), called Hamilton – Jacobi equation. This is an equation with partial derivatives.



Reciprocally, given the equation with partial derivatives (4.2.4), we have: ,

cu



4.3 THE LAGRANGIAN VARIANT

Given an ordinary differential equation system, of irst order, written, for example, in cinematic form, There is an integrant factor , by the multiplication with which, the transforms into one that is equivalent to it, auto adjunct: ( 4.3.1 )

.

To this a Lagrange function is associated property that Euler-Lagrange functions are:

,

. The Lagrange function

, has the property that the action function:

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, with the

enjoys the property that

, but

(4.3.2) . By identifying (4.2.4 ) with (4.2.5 ), it results:





Consequently:

and therefore



or (4.3.3)

.

CAP V ROUTHIAN FORMALISM § 5.1 DEFINITION OF ROUTH’S FUNCTION It is known [26] the way to introduce Routh’s function, and, also, of the formalism bearing the same name. In the following, we will handle the study of the reverse problem of Newtonian mechanics, in this formalism; we will consider the evolution equations of a dynamic system, written in Newtonian form and we will look for the necessary and suf icient conditions of a Routh function, so that the evolution equations can be put under Routhian form, equivalent to Newton’s equations. This formalism is a hybrid of Lagrangian and Hamiltonian formalism. 1. De inition of Routh’s function Given the Newtonian equation system, written under the differential form:

(5.1.1)



equations written on , where M is the con iguration variety. Let us assume that, on the M variety, a differential system (distribution), completely integrable, and whose dimension is n-r. Therefore, our problem has a global character, to the extent that such a system exists globally. We will say that a local map is adapted to the F system if its irst coordinates belong to the latter. In any point

, it is de ined, by means of the F system, the vector space

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, so

that (

) is a subbundle of the tangent bundle. At the same time the annihilator F0 is de ined, in

r size, and with it a subbundle of the cotangent bundle (

), for which

is formed

of all the forms de ined on Tx, null on Fx. this being given, the bundle ( ) is built and the Routh function is de ined as being an arbitrary function, differentiable and with real values: R ; in the (

) adapted map: R:

R

,

, so that the following information is veri ied:

(5.1.2) and the systems (5.1.1) and (5.1.2) are equivalent. Solving this problem is done by de ining an application



,

function which we will assume irreversible in relation with : , . The equivalence of the systems (1) and (2) will be done with the help of an integrant factor, by means of the formula:

(5.1.3)



cu

and with the condition . From the hypothesis of existence of the functions φ, g, h and R, so that the system (5.1.2) to come from (5.1.1) by means of the formula (5.1.3), the following system is deduced: ,

, . These conditions are explicitly written under the form



,

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dt. Identifying the coef icients , and dt, and their convenient grouping, leads to the following relationships written in the “uncoupled”:

(5.1.4) and, respectively, the consequences:



(5.1.4’) relationships, that, together, can be put under the form:



(5.1.4’’)



The last relationship is: ------------------------------------------------

In the calculation it is used the commutativity property of the image reciprocal to the differentiation (

).

(5.1.4’’’) As it can be seen, in the formulas (5.1.4’’) the expression was highlighted (based on the equations (5.1.2)):

therefore it is imposed to consider „Lagrange’s function”:

(5.1.5)

):



with the help of which the irst equations are written (5.1.5) in relation to qi the following relationship is deduced (5.1.6) Integrability conditions

.

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by deriving

a) Let us now impose to the (5.1.4) system the integrability conditions

(5.1.7)



Considering (5.1.6), we have:

therefore

formula that leads, together with (5.1.7),

to

the last relationship in (5.1.7) leads to: Summing up, the integrability conditions are obtained:



(5.1.7’) b) A part of the irst conditions (5.1.7) results from s the equations (5.1.5.a), which, written conveniently under the form: implies

;

from these last relationships it is deduced that the matrix maintained also for its reverse

is symmetrical, property that is

, therefore:

The irst relationships (4) lead to:

.

c) Let us derive the relationship (5.1.4’’’) in relationship with

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, we get

;

the last term transforms as follows:

and with

the following is obtained

, because

. Therefore:

so the following formula is reached: With the Lagrangian notation (5.1.5), the formula (5.1.8) is written:

.

, Relationship from which the Euler- Lagrange equations are deduced:

, not considering a 1-form

, dependent on time and position.

By doing in the formula (5.1.8)

, the following relationship is obtained:

(5.1.8’) from which it results

,

Statement 1 A system of functions , constitutes the components of a partial impulse function if they form a set of solutions for the equation (5.1.8’). The relationships (5.1.8’), written in developed form, lead together with (5.1.4) to a part of the integrability conditions (5.1.7’) (considered for ) c) Other compatibility conditions may be obtained by deriving the relationship (5.1.43), related to

, by changing the indices

and

between them and the subtraction, etc.

§ 5.2 Applications 1. Find the Routh function of a symmetrical peg top in an external ield the cyclic coordinate (ψ,φ,ɵ being Euler’s angles). Solution Lagrange’s function is considered:

eliminating

,

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Routh’s function is : ; the irst term in this expression is a constant that may be omitted. 2. For the dynamic system, that corresponds to the igure below, [5], the coordinates of the mass point m and the projections of the forces are given by



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where ro is the r distance for which the k arc is free. The evolution equations (in Newtonian form) of this system are written according to the curvilinear (generalized) coordinates under the form:



Let us look for a solution of the system (14) ? of the form:

:

from where: . From (4) it results For the other functions g the following equations are obtained:

.

from which it results

,

,

and, therefore, the following values are obtained: . For the h functions the following values are obtained: ,

,

, therefore

from where



. From (5.1.4) it results that

,

, therefore

and, in the end, . § 5.3 LAGRANGE’S FORM OF A DYNAMIC ROUTHIAN SYSTEM Let us consider the evolution equations, obtained from (1) by the multiplication of an integrant factor [3]

(5.3.1) to it Lagrange’s 2-form is associated:



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(5.3.2) To the equations (2) Lagrange’s form corresponds:



(5.3.3)



The two forms

integrant factor.

.

. and

admit the same characteristics if and only if

, considering that a potential proportionality factor may be included in the

The explicit factor of the formula

leads us to:

. By identifying the coef icients of the forms system results:

and

, the following equations

, -

,

These equations are equivalent to the following consequences:



(5.3.4) Statement 5.3.2 The relationships (12) are consequences of the relationships (5.1.4) and reciprocally (5.1.4) are deduced from (5.1.12) disregarding the adding of a symmetrical tensor hij (nonsigni icant).

Statement 5.3.3 The form is closed. Indeed, it is (locally) an external differential of the 1-form

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(5.3.5) Compatibility conditions

.

Based on statement 3, it results that the form is closed, from where it is deduced that must be closed (it satis ies Maxwell’s principle). From this condition, it results the following equations system [3]:

(5.3.6)



These conditions coincide with the existence conditions of an integrant factor that leads to the existence of Lagrangians. Statement 5.3.4 A system of functions (g,h) are the components of an integrant factor, which ensures the existence of a Routh function, if and only if they are solutions of the system (5.3.6). ``

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CAP. VI BIRKHOFFIAN FORMALISM OF CLASSICAL MECHANICS As we have seen in previous chapters, in own dynamical systems (which we will continue to handle), classical Lagrangian systems coincide with classical Hamiltonian systems. Assuming that a Lagrange or Hamilton function cannot be determined (it is not known or it does not exist), we seek to generalize them so that the motion equations given in the Newtonian form can be obtained, for example, under a generalized Hamiltonian form, so that they also come from a variational principle. Such generalization was given by Birkhoff [6] and presented by Santilli [50]. It is called the Birkhof ian generalization of Hamiltonian mechanics. For its description, we will start from the de inition and study of Hamiltonian formalism of classical mechanics. In paragraph 6.1, the Newtonian equations are transcribed in the Birkhof ian form and are interpreted. Paragraph 6.2 completes the relationship between these two types of dynamic systems (Newtonian and Birkhof ian). Paragraph 6.3 formulates and analyses the reverse problem of classical mechanics in Birkhof ian formalism, and paragraph 6.4 considers the second order Lagrangians and solves the direct problem, this time more dif icult. It shows under what conditions a second order Lagrangian leads to the EulerLagrange equations of the third order, that are linear combinations of Newtonian equations and their derivatives. At the end, we describe Noether's theorem in this case.

§ 6.1 BIRKHOFFIAN DYNAMICS

6.1.1 The evolution equations of a dynamical system, written under Newton’s form, are differential equations of order two on M. Let us assume that we do not start from a Lagrangian and let us transcribe Newton’s equations under the form of a irst order equation system on T*M; we will de ined the application ϕ:(t,q,v)∈R×TM→ (t,q,p)∈ R×T*M, where the functions pi=ϕi(t,q,v) have the property: det( ) ≠ 0; it results that: (6.1.1) vi = ψi(t,q,p). From Newton’s equation we have: -Fi(t,q,ψ) = 0,

and therefore:

formula from which:

i-(

(6.1.2) ( We therefore have:

)ψ[

), or:

i-

]:=χh(t,q,p).

(6.1.3) vi = ψi(t,q,p), χi(t,q,p) = ym+1 , system that can also be written under the form: (6.1.4) if it is denoted:







Φμ = The (6.1.4) equations represent the countervariant form of Newton’s equations. 6.1.2 Classi ication of Birkhof ian dynamic systems

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186

Let us now assume as given, on the phases space, the presymplectic 2-form Ω = Ωμνdaμ∧daν, with the help of which the covariant normal form corresponding to (6.1.4) can be written: (6.1.5) Ωμν Φμ=ΩμνΦν. De inition 6.1.1 A system of irst order differential equations, de ined on the cotangent variety, constitutes Birkhoff’s equations of a dynamic system, if there is a function B = B(t,a), called the birkhof ian of the system and a presymplectic 2-form =Rμdaμ, and the system can be written under the form:

Ωμνdaμ∧daν closed, so that locally Ω = dθ, with θ

(6.1.6) Observation 6.1.2 Hamilton’s equations can be obtained by the Birkhof ian representation for y

=p=

, R = Ro = (p,0), B = H(t,q,p). The following is obtained: Ωμν De inition 6.1.3 The (6.1.6) system is called autonomous if the functions Rν and B do not depend



explicitly on time. Such a system is written: Ωμν(a)



The tensor Ωμν = is called Birkhoff’s tensor. These systems also bear the name of generalized Hamiltonian systems. The (6.1.6) system is called semiautonomous if the R ield does not depend on t. In this case it results: Ωμν(a) The (6.1.6) system is called nonautonomous if its equations are of the (6.1.6) form:



Ωμν(t,a) A birkhof ian system is called regulated if det(Ωμν)≠0 and singular if det(Ωμν)=0. In the case of a regulated birkhof ian system, it can also be written under the countervariant form:

or explicitly:



. 6.1.3 The analytical interpretation of Birkhoff’s equations Let us consider the pfaf ian action:



(6.1.7)



whose variation is, with δ =

A=

:

δA =

=



f

f

f

f

f

f

f

f

f

f

187

= and which, by cancellation, leads to Birkhoff’s equations. De ined as above, Birkhoff’s equations derive from a variational principle.

The (6.1.7) action of the system is therefore of the form:

A

=

where the

Lagrangian is of a special form (linear in ), totally degenerate and de ined on R×T*M. The characteristic (extremal) curves associated to it, on T*M, are given as solutions to the Euler-Lagrange equations: 6.1.4 The algebraic interpretation of Birkhoff’s equations Let us consider a function A=A(a) and calculate the total differential along the solutions to Birkhoff’s equations. We will obtain:

(6.1.8) In this formula and below [ , ]* represents the generalised Poisson bracket (built with Ωμν). Let us observe the multitude of functions de ined on T*(M), with the opration [ , ]*, constitute a Lie algebra. Indeed, considering the properties: Ωμν+Ωνμ=0, ,

the axioms of the Lie algebra are checked:

[A,B]*+[B,A]*=0, [[A,B]*,C]*+[[B,C]*, A]*+[[C,A]*,B]*=0. 6.1.5 Geometrical interpretation of Birkhoff’s equations In the case of autonomous and semiautonomous systems the 1- form (generalized Liouville) can be considered and with it the (symplectic) 2-form

. In view of studying the non-autonomous systems we will consider on R×T*(M) the local coordinates and vector: (6.1.9) , , with the help of which we will build the 1-form (the integral invariant of Poincaré-Cartan): (6.1.10) and its exterior differential (the presymplectic form of space):



More explicitly, the matrix

is written:

f

f

f

188

. The characteristics of the receive the form:

2-form are obtained as solutions to Birkhoff’s equations, which

(6.1.11) or more explicitly:

,

, . The last 2n equations are exactly Birkhoff’s equations and are under the form:



(6.1.12) . The irst relationship is identically satis ied, based on the others; it is therefore a consequence of these and constitutes a fundamental theorem of birkhof ian mechanics. Let us write the autoadjunction conditions of the system (6.1.12): (6.1.13) , , . They are veri ied by the birkhof ian systems and reciprocally. Statement 6.1 The necessary and suf icient condition for a irst-order equations system, written under the general form (6.1.12), analytical and regulated in a star-shape ield from R T*(M), to be autoadjunct is that he is birkhof ian. Observation 6.3 An autonomous (semiautonomous, non-autonomous) birkhof ian system de ined in the T*(M) variety, equipped with the symplectic structure de ined b the Ω 2-form, is given by a B function and by an X vector ield, which veri ies the relationship iXΩ=dB. 6.1.6 Interpreting Birkhoff’s equations on the tangent bundle Given the variety V=R T*(M) and on it the presymplectic 2-form (6.1.14) To it we associate the characteristic system:



(6.1.15) . Let us now consider the variety W=R T(M) and the function

given by the formulas:



With the notations:

The reciprocal image through

,

, we have

of the

.

2-form is:

(6.1.16)

,

and the reciprocal images of the

forms are the forms:



(6.1.17)



.



The following system is attached to the form

:

f

f

f

f

f

f

f

f

f

f

f

f

f

f

189



(6.1.18)



.

Based on the hypothesis that the matrix (6.1.17) and (6.1.18) are equivalent because:

is of rank 2m+1, it results that the systems

;

the commutative diagram takes place:





. Conclusion On the tangent space, produced through R, Birkhoff’s equations describe a Newtonian dynamic system.



For writing Newton’s equations, we can proceed as follows: the system

Cramer system with 2m equations and 2m unknowns

, the speeds

is a are expressed and

the their values are inserted in , from where it results: . We call the direct problem in Newtonian mechanics, in birkhof ian formalism, giving the 1-form potential , determining the presymplectic structure , of the characteristic equations and deducing Newtons equations (of the force ield that governs the evolution of the system), it results that through the above considerations the direct problem of Newtonian mechanics in the birkhof ian formalism is entirely solved. We shall name dynamical systems of such type the N.B. system (Newton-Birkhoff).

Example 6.1.2 Given the potential

and the birkhof ian .

Birkhoff tensor is:

and it is regulated where

. Birkhoff’s equations are:

f

f

f

f

190

Considering the function form:

, we obtain the motion equations under Newton’s

.

§ 6.2 BIRKHOFFIAN DYNAMIC SYSTEMS

6.2.1 The relationship between birkhof ian and Newtonian systems We now endeavour to show to what extent Birkhoff’s equations generalize Hamilton’s and which is the report between the former and Newton’s equations. The following Pfaf ian is associated to a birkhof ian system: ,



and together with it the Pfaf ian action on T

:

A=



Considering that

the action on A is transcribed on

:

A

=



=: By cancelling the variation of the action A (δA = 0), we obtain the following results:



(6.2.1)



The action of the system now being given on

through the Lagrangian:

u where:

(6.2.2)





For the explicit calculation of the equations (6.2.2) we have:





f

f

f

f

191



(6.2.3)



and therefore:

. From these considerations we deduce that an N.B. system of on the M variety, a second order variational system. Take c a curve on TM and its image through j on T*M. The actions pe T*M and on TM are equal (based on the invariance of the curvilinear integral to diffeomorphisms): A

If we change the potential

the action changes according to the formula:

formula from which it is deduced that their variations cancel each other simultaneously , property that is kept also through the application

on TM:

and therefore

from where it results that the (6.2.3) equations are independent of a gauge transformation of the potential form .

6.2.2 The dynamic systems Birkhoff-Santilli As it results from the form of the (6.2.3) system, it is of irst order, if and only if:



(6.2.4)

,

formula from which we deduce that the Vi functions are the partial derivatives in relation to function: Based on the relationships (6.2.2), the condition (6.2.4) becomes:



and it is therefore realised if: (6.2.5)



192

f



, of a



Reciprocally, with the hypothesis det



Indeed, from the relationship

the condition (6.2.5) implies (6.2.4) . the relationships

are obtained

where , therefore: ; Consequently, we have: Statement 6.2.1 A necessary and suf icient condition so that an N.B. system to be of type B.S. is for a function

to exist so that

A condition equivalent to the irst one is to have an

S=S(t,q, ) function so that Birkhoff’s tensor for such a system is:

and the motion equations of the system are reduced to: and can be obtained from the ( irst order) variational principle applied on T(M).

6.2.3 The canonical form of the B.S system



The presymplectic 2-form



being replaced by exact local hypothesis is, locally, the the

differential of a 1-form potential , by meeting the conditions (6.2.4). This is however de ined until the local differential of a function S(t,q,y), therefore, if a gauge transformation is done the following potential is obtained remains unchanged, leads to the same trajectories. We will have:

form with which

We will be able to choose the S function in such a way that system is completely integrable, based on the relationships (6.2.5) and therefore:

because this

. In the hypothesis of such gauging, the form

becomes:

so that:

and the evolution equations take the form:

, which tells us that the B.S. systems are Lagrangian. Statement 6.2.2 The necessary and suf icient condition for a birkhof ian to be of B.S. type is that it admits a (generalized) semi basic Liouville form:

f

f

f

f

f

f

193



, Birkhoff ‘s equations are reduced in this case to the form: ,

or to the differential form:

,

,



(6.2.6)



Take the Newtonian system, given by the equations:



Let us assume an integrant factor of the form:

.



with which, by multiplication, the motion equations are put under the equivalent form:

(6.2.7)





These equations must coincide with the reciprocal image of the equations (6.2.6) through j, which are written: (6.2.6') . By identifying the coef icients of the two systems (6.2.6') and (6.2.7) the connection relationships are deduced:

,



,

from where:

.

194

f



§ 6.3 THE REVERSE PROBLEM OF CLASSICAL MECHANICS IN BIRKHOFFIAN FORMALISM

6.3.1 Let us assume as given a dynamic system through Newton’s equations:

and with them, an integrant factor (with non-null determinant):

with which we are led to the equations:

.

Given the Legendre transformation through

with its reverse



we will have on R T*(M), the equations:

,

,

system that is of the form: , ,

or with the notation change: ,

of the form:

(6.3.1) To the system (6.3.1), if the autoadjunction conditions are imposed, the closed 2-form is associated: whose characteristics are exactly the trajectories of the given dynamic system. As it is known, the necessary and suf icient condition for the 2-form to be closed, is for the system (6.3.1) to be autoadjunct, condition equivalent to the property that the system (6.2.7) is integrable. Based on the reciprocal of the Poincaré lemma we will obtain the 1-form potential whose coef icients are given, for example, by the formulas: , where



.

The reciprocal image of the

form, by the impulse application j, leads to the 1-form:

f

195

f



,

whose extremals will be given by the formulas:

; these constitute a system of equations, which admits among its solutions the solutions of the given system as well. For the practical realization of the reverse problem it may be considered as a Legendre transformation, the simplest transformation ph=vh (eventual ph=mvh) so that the reciprocal images through of the evolution equations become: Let us now impose on the system (6.3.1) the integrability conditions (6.1.13), among which the



irst tell us that from the coef icients

and

only the following are independent

Let us look for a solution to the system (6.1.13) under the form Cab = Cab(t). Thus, the second group of equations (6.1.13) is identically veri ied. The last compatibility equations are in this case also n(2n-1) and they are reduced to a system of linear and homogenous equations, with variable coef icients: but, as it is well known, such a system always has solutions.



If the force ield is under the form: constant, we have constant coef icients:

with the coef icients

,





.

f

f

f

f

196

f

and

and therefore the system becomes a linear system with

,

f

f



and, together with it to the Lagrangian:

Such a system always has solutions under the form , where g is a square root of the characteristic equation associated to the system, and gi are for example minors extracted from the coef icient matrix (assumed to be of n(2n-1)-1 rank) putting aside a column. Example 6.3.1 Let us consider Whittaker’s system, which as it is known, is not a classic Lagrangian system:



As a linear system, it is written:



Let us consider the integrant factor:

with the help of which, by multiplication, the given system becomes:



Let us consider the impulse function

with which it results:

This system is of the form:

and it has to be auto-adjunct (3.2.66).

The irst conditions:

give



therefore:

Assuming the constant integrant factor, the second group of relations is ful illed. The third one results in:

meaning:

f

f

f

197

where , The system becomes:





With the matrixes:

,

We build the 2-form:

.



Therefore:







from where:

and therefore:

leads to the evolution equations, corresponding to Whittaker’s equations. To determine the functions Ra and B, the following system of equations can be written:

198

This system being integrable (based on the auto-adjunction conditions) is integrated and it is shown that it allows as solution: 6.3.2 The Birkhof ian representation theorem of classical mechanics Let us now show that any Newtonian system admits a Birkhof ian representation and, as such, it comes from a variational principle. In this purpose, we will prove: Theorem 6.3.2 (universal one, of Birkhof ian representation). Any Newtonian system admits locally a Birkhof ian representation. Proof Taken the Newtonian system system on the tangent variety

(6.3.2)



which we shall write as a irst order



Let us consider the application system on the cotangent variety:



, with the help of which the system becomes a

(6.3.2') This system can be written concentratedly under the form:

(6.3.2") if the following notations are used:

,

. Proving the universal theorem lies in showing that such a system is always equivalent with a Birkhoff type system:



(6.3.3)



,

equivalence from which we propose to determine the functions

Let us eliminate between (6.3.2") and (6.3.3) the derivatives

and B. ; we will obtain:

.

This is a system with 2n equations with partial derivatives with n+1 unknown functions, through

whose solution the potential is obtained and the Birkhof ian function B, functions A being arbitrary (under the condition that they are derivable). The following cases are noticed:

f

f

f

f

f

f

199

Case 1 Let us look for a non-autonomous solution, the then the equation system can be explained as follows :

functions depend explicitly on time,

and it is therefore in the situation where Cauchy-Kovalevskaia’s existence theory applies, which ensures the solution and therefore the proof of the theorem. Case 2 Let us look for a semi-autonomous or potentially autonomous solution (if the force ield is stationary). In this case the for example form:

functions do not depend explicitly on time. We will ix a coordinate,

and we will explicitly transcribe the equation system with partial derivatives under the

, where it was considered that B= . This latter system is also within the applicability conditions of the Cauchy-Kovalevskaia theorem and, as such, it admits a solution also in this case. With this we consider the theorem as proven. § 6.4 SECOND ORDER LAGRANGIANS In this paragraph we undertake, on the one hand to formulate a direct way of constructing Lagrangians – which also depend on accelerations – that correspond to a Newtonian dynamic system given apriori (the reverse problem), without using the Birkhof ian interpretation of the system, and on the other hand, let us research the direct problem, in which we give a Lagrange function and we look for Newton type equations, to which the given function corresponds as a Lagrangian. 6.4.1 Autoadjunct variational systems of the third order. Autoadjunction conditions. General form of third order equations Let us consider a system of n ordinary differential equations, of the third order, written under implicit form:

(6.4.1)



=0, i=

,

for which we assume as met the existence conditions of the implicit functions in relation with the variables:

(6.4.2) det i,j= . Given I=[t 1, t 2 ] a compact interval and the differential path c: I

M(q); the

lift corresponds to it where J3(M) is the space of the

third order jets. Let us associate to the path c a “variation” of it C : a differentiable application C : , with the property C (0,t)=c(t), J being an open interval that contains the origin. A C variation may be interpreted as a uniparametric family of paths, called admissible along which we will consider the functions F i. We denote by: P= {C = C(e,t), eÎJ, tÎI} these variation’s set. Take now: a lift of C. Let us de ine a “virtual variation” of the admissible paths, through:

f

f

f

f

200

(6.4.3) i= Let us denote with Mi(h), the system of forms with variations associated to the (6.4.1) equations, expressions deduced by considering the F i functions along the paths of the P family and let us calculate their derivatives in relation to e for e = 0:



(6.4.4) in which, for e = 0, it was denoted:

,

(6.4.5)



The virtual variations

de ined through (6.4.3) are not unique, they depend on the C

variations choden from the P family. For another admissible variation:





Generally the forms with variations are written under the form:

(6.4.6)

variation we will have the virtual

i=

,

with whose help the form system with variations can be written:

(6.4.7)

.



De inition 6.4.1 Two form systems with variations

functions of t) are called adjunct if there is a function

(6.4.8)

and

(with the coef icients

so that:

Statement 6.4.2 For any form system with variations M i, there is a unique form system with variations adjunct to the given system. Proof The existence is deduced by direct calculation, as follows:

(6.4.9) Therefore, it results the existence of the adjunct system with variations:

f

f

f

201

(6.4.10)





(6.4.11)





To prove the uniqueness, let us assume that there are two adjunct systems: , and two functions



and

and

, respectively that check the relationships (6.4.8):

(6.4.12) . By making the difference of the relationships (6.4.12), we obtain:

(6.4.13) relationship that is deduced by integration:

,

(6.4.14) , regardless of what might be the virtual variation h, so that: (şi J' = J''). De inition 6.4.3 A form system with variations Mi(h) is called autoadjunct if it coincides with its adjunct system:

(6.4.15) By identifying the relations (6.4.4) and (6.4.7) , we obtain the following relations:





(6.4.16)

,

, ,

, which represent the autoadjunction conditions of the (6.4.4) forms with variations. The system (6.4.1) is called autoadjunct if the system of its forms with variations is autoadjunct. By using the notations (6.4.5) in the forms (6.4.16), we obtain for (6.4.1) the autoadjunction conditions set:



,



(6.4.17)

,

,



.

6.4.2 The main form of third order equations Let us now consider the case of the systems written under the main form: (6.4.18) . For such systems, the autoadjunction conditions are transcribed under the form:

f

202



(6.4.19)

Aij + Aji = 0,

,

,

Considering the linearity in the superior order derivatives, the (6.4.19) conditions can be put under equivalent form:





(6.4.20)







The Aij functions do not depend on

.



6.4.3 The canonical form of third order equations

it:

Given a second order Lagrangian L=L

. The Euler-Lagrange equations are associated to

(6.4.21) , which represents a fourth order system. If, in particular, a “special” Lagrangian is considered under the form: (6.4.22) L then the (6.4.21) associated Euler – Lagrange equations represent a third order system, written in a main form (6.4.18) in which:

(6.4.23)



. Statement 6.4.4 The (6.4.21) Euler-Lagrange equations written for an L function of the form (6.4.22) are autoadjunct.

203

Proof These equations are of the form (6.4.18) in which the coef icients Aij and Bi are mentioned in (6.4.23). From their de inition in immediately results the satisfaction of the relationships (6.4.201), (6.4.202), (6.4.203), (6.4.204) and (6.4.205). To prove the relationship (6.4.206), each of the parts of the equation are calculated and the same expression is obtained. In verifying the relationship (6.4.207) we consider (6.4.206), (6.4.204) and (6.4.203). The last of the relationships (6.4.20) is checked based on relationships (6.4.206), (6.4.202) and (6.4.204). From statement 6.4.2 the following results Corollary A necessary condition for the third order differential equations of a form system (6.4.18) to be Euler-Lagrange equations of the type (6.4.21) written for a Lagrangian of the form (6.4.22) is for a system to be autoadjunct. Theorem 6.4.5 (fundamental analytical). The necessary and suf icient condition for a third order equations system, written in the main form: (6.4.24) to admit, in a star domain, a direct analytical representation under the Lagrange form: with a function of the form L= , is for it to be autoadjunct. Proof The necessity of the autoadjunction was proven in statement 6.4.2. To prove the suf iciency, we will assume that the system (6.4.18) is autoadjunct, so Aij and B i which are known, verify the conditions (6.4.20). From (6.4.23) we will determine the functions Vi and W so that the Lagrangian (6.4.22) will be completely determined. Considering (6.4.231) mentioning that t and q are considered parameters [51], we build tge exterior 2-form with the antisymmetric Aij coef icients: . This 2-form is closed: dw = 0 based on the relationships (6.4.201), (6.4.202) and (6.4.203). Consequently, there is a 1-form so that . In this case, according to the Poincaré lemma, it can be integrated in the relationship (6.4.231), from where we obtain ( 6.4.25) To determine the W function, we consider the relationship (6.4.232) which we also write under the form:

(6.4.26)

.



In the last relationship, we derive related to

and we obtain:



(6.4.27) We replace (6.4.27) and (6.4.26) and we obtain:



(6.4.28) Observations 6.4.6 a) The relationship (6.4.27) is symmetrical, therefore it can be integrated.



f

f

f

f

f

204



b) The left side of the relationship (6.4.27) does not depend on

(6.4.203), (6.4.204) and

. Using (6.4.201), (6.4.202),

(6.4.205) it can be shown, by derivation, in relationship to

, that

.

c) Through the same procedure it can be shown that

using also the identity

(6.4.29) . We will show that if the conditions (6.4.20) are met the system formed from (6.4.27) and (6.4.28) is integrable, therefore the W can be determined.

In [47] is was shown that the system (6.4.27) – (6.4.28) is integrable if the following conditions are met: ,



(6.4.30)



The relationship (6.4.301) results from (6.4.201,3,4). Similarly, from (6.4.201-5) and (6.4.301) it results (6.4.302). From the relationships (6.4.201-6), (6.4.301,2) and after simpli ications, it results (6.4.303). For the relationship (6.4.304) (6.4.20), (6.4.29), (6.4.301-3), b) and c) are used. In [47] it is shown that in the conditions (6.4.30), W is given by: (6.4.31) where:

W

, ,



(6.4.32)

,

while Zij and Xi are given by the formulas:

,



(6.4.33)





It results that the theorem is completely demonstrated.



6.4.4 The reverse problem of classical mechanics Take the evolution equations of a dynamic system, written under the Newtonian form m(i)



, or by division by m(i), under cinematic form: (6.4.34) As it is known [47], this system always admits a Lagrange function under the form (6.4.22) with the property that Euler-Lagrange equations (6.4.21) admit as solutions the solutions of (6.4.34). The equations under the form (6.4.21) written for an L function of the form (6.4.22) are differential equations of the third order. They admit as solutions the solutions of (6.4.34) if they are linear combinations of (6.4.34) and their derivatives. Meaning, there is an integrating factor (Aij, gij) so that:

f

205

(6.4.35) is of the form (6.4.21). For this to happen, it is necessary and suf icient (theorem 6.4.1) to be veri ied the autoadjunction conditions (6.4.20), for which:



(6.4.36) Therefore, the following takes place:



Statement 6.4.7 The force ield (accelerations) being given, the necessary and suf icient conditions for the integrant factor (A,g) to put the equations (6.4.35) under the form (6.4.21) are:

(6.4.37)











Proof It is considered in the autoadjunction conditions (6.4.20) the functions Aij, given by the integrant factor and Bi de ined by (6.4.36) with the help of functions Aij and gij. The irst three conditions in (6.4.20) remain unchanged. The other conditions (6.4.37) are obtained by replacement. By identifying the left parts in the equations (6.4.21) and (6.4.35) for which we assume as met the autoadjunction conditions (6.4.37), leads to the system: (6.4.38)





f

f

f

f

f

f

206



Statement 6.4.8

The functions Vi and W and together with them Lagrange’s function

are obtained by means of the formulas (6.4.25) and (6.4.31), respectively, and in the coef icients given by the formulas (6.4.32) and (6.4.33), and are given by (6.4.38). Proof It results by applying theorem 6.4.1. Example 6.4.9 Take the dynamic system with two degrees of freedom described through the Newtonian evolution equations:



We look in (6.4.37) for an integrant factor of the form Aij=const. , gij=const. A12 + A21 = 0, g12 – g21 + A11 + 2A22 + A12 = 0, 2g12 – A11 –2A12 = 0, g21 + g22 – 3A12 + A22 = 0, g11 + g12 – 2g22 + 3A11 + 2A22 = 0, 3g11 + g12 – g21 – 2g22 = 0. The system admits the solution: g11=g12=g22=1, g21=2, A11=A22=0, A12=-A21=1. With this integrant factor, the (6.4.38) system is written: ,

, ,







and admits as solution the functions: , ,

and with them, Lagrange’s function:

f

207





Euler-Lagrange’s equations (6.4.21), become: ,



,

Observation 6.4.10 Using the formula (6.4.25) it results: ,

, ,

and correspondingly, Lagrange’s function:



,





Example 6.4.11 Thake the system with three degrees if freedom:

, . For this system we will look for a constant integrant factor (Aij, gij) and we will ind:

, -A32 = 1 It results for the functions Vi and W the values: ,

,

,

A12 = -A21 = A13 = - A31 = A23 =

,





And for the Lagrangian, the expression:



equations have the expression:

Euler-Lagrange’s

, ,

.

f

208



6.4.5 The direct problem of classical mechanics Formulating and solving the direct problem of classical mechanics, in the most general case,

consists in providing a (generalized) Lagrange function L=L( ) of the form (6.4.22) and deducing (from it) a system of Newton equations of the form (6.4.34), that come from applying the minimum action principle (under a general form). With each generalized L Lagrangian of the form (6.4.22) an antisymmetric components’ matrix is associated.

(6.4.39) . De inition 6.4.12 A generalized Lagrange function (6.4.22) is called ordinary (non-singular or nondegenerate) if:



(6.4.40) We will say that L is singular otherwise. Theorem 6.4.13 A Newtonian system (6.4.34) corresponds to any non-singular Lagrangian (6.4.22). Proof Take the Lagrangian of the form (6.4.22) to which the (third order) Euler-Lagrange (6.4.21) are associated; let us assume there is a Newtonian system (6.4.34) and an integrating factor (

) so that:



By identifying the coef icients, the following relations are deduced:



a)



b)

.



c) The formulas a) de ine the functions Aij, while the formulas b) and c) in a number of n(n+1) de ine the functions gij and f i . If gij given by the expression b) are introduced in c) relations are obtained only in the unknown conditions f i and their derivates:





These relations can be written under the form:

d) where:



f

f

f

f

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:

e)

,

The scalar functions can be determined by solving the system e) as the functions Aik are known. It therefore results that the system d) is compatible if the following relations are also veri ied:

f)



obtained by multiplying the relations d) by These relations are of the form:

, adding and considering e).



g)



If 2p=n, then the system d) is irreversible and therefore we can develop all the derivates

(or



for a ixed i) and therefore are met the conditions in Cauchy’s theorem for local existence of

the solutions for the d) system in

. If 2p=n-1, then the conditions g) are reduced to only one which

supplies a function with the help of all the others. Once this is introduced, together with its derivates, in d) it leads to the same results, with which the theorem is fully demonstrated. More generally, if 2p