Dynamic Differential Systems [II, 1 ed.] 9789730395532

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Prof. Dr. VIRGIL OBĂDEANU

DYNAMIC DIFFERENTIAL SYSTEMS PART ONE FIELDS AND WAVES ASSOCIATED TO THE DYNAMIC SYSTEMS PART TWO GEOMETRIC STRUCTURES ASSOCIATED TO THE DYNAMIC SYSTEMS

VOLUME II Translator Ciupa Anca

Timișoara 2023

ISBN 978-973-0-39553-2

Prof. Dr. VIRGIL OBĂDEANU

DYNAMIC DIFFERENTIAL SYSTEMS PART ONE FIELDS AND WAVES ASSOCIATED TO THE DYNAMIC SYSTEMS PART TWO GEOMETRIC STRUCTURES ASSOCIATED TO THE DYNAMIC SYSTEMS

VOLUME II Translator Ciupa Anca

Timișoara 2023

Translator Ciupa Anca Copyright © 2023Flavius Ungureanu

f

E-mail: ungureanu [email protected]

THE ORIGINS OF VARIATIONAL CALCULUS AND ITS HISTORICAL EVOLUTION As it is well known, from the remotest times in history, mathematicians were interested in the problem of maximization and minimization. Thus, in the first century A.D., the Greek mathematician and philosopher Heron of Alexandria already noticed that the path travelled by a ray of light during its reflexion on a plain mirror is minimum. Old Heron said then, as a principle, that the ray of light follows the shortest path and demonstrated thus the law of reflexion: the angle of incidence and the angle of reflexion are equal. At the end of the 17th century and beginning of the 18th century, the calculus of variations was truly expressed. The French mathematician Fermat, predecessor of the differential and integral calculus, amended a rule of determining the function minimum calculus. In 1657 he expressed his general principle of geometrical optics, according to which, in order to move from one point to another, light follows the path (road) of minimum or maximum duration. Sometime later, in 1661 he discovered that this principle is equal to the laws of Descartes on reflexion and refraction. Yet, no systematic method was established before the emergence of mathematical analysis. Fermat is one of the main predecessors, though Newton was at the origin of a coherent theory of the evanescent calculus. Leibnitz, inspired by Pascal’s infinitesimals, arrived, sometime later (1775) to the same synthesis, but presented under a different forms and soul. The swiss mathematician Euler, disciple of Bernoulli, became the main founder of the calculus of variations, publishing the equation, that bears his name, in 1736, as well as a much more comprehensive method in 1744. A while later, the French mathematician LaGrange introduced the notion of variation and gave the modern shapes of the demonstrations of Euler’s results that he considerably perfected. His methods, purely analytical, allowed him to constitute the calculus of variations as an autonomous branch of the evanescent calculus. Sometime later, LaGrange specified the principle of minimum action expressed by Mauperthuis in 1774. He applied these results to rational mechanics and obtained the equations that bear his name: in the mechanics of fluids we also find a new demonstration of the Euler equations, in this field. In the 19th century, both mechanics as well as the theory on the nature of light lead the Irish mathematician, Hamilton, to the mathematical and deductive theory on optics. Thus, we see the progress of variations in solving differential equations. In this context of ideas, this material is trying to fit somehow, following exactly the principles and methods mentioned above, in expressing and studying certain evolution problems encountered in biology and economy.

FOREWORD. DYNAMICS OF MICROPARTICLES Starting from the study of the properties of light, that is its undulatory nature and that of having impulse, Albert Einstein suggested the idea that the wave of light propagation must be accompanied by a particle, which was later called photon. Subsequently, in the study of the properties of the electron, considered as a particle, de Broglie associated it with a nature. Probably considering the unity of the world. Let's reformulate, in summary, the presentation of this theory. It is known that by the 1900s, physics consisted of two branches, important today too. 1





INTRODUCTION

p = mv = h λ , where λ is the distance between two expression of the corpuscle impulse: consecutive maximums of the wave, of the wavelength. λ= h p , fundamental relationship of the theory. De Broglie generalized this Thus relationship for the case where the particles move in a force field, which derives from a potential so λ and p vary from a point to another in space. λ ( x, y , z ) = h ( x, y , z ) p that is . This parallelism between corpuscle and wave, allows de Broglie to identify Fermat's principle from the optics with respect to light, with Maurthuis’s principle from the classical mechanics of

2

























One of these branches was the physics of substance (the physics of amorphous matter), based on the concept of corpuscles and atoms, supposed to comply with the classical laws of Newtonian mechanics, and on the other hand, of radiation physics (undulatory physics), founded on the concept of wave propagation in a hypothetical continuous environment, called the ether of light, or the electromagnetic ether. Thus, it was necessary to present for the light two contradictory theories, undulatory, as well as corpuscular, based on the property of its propagation, and the corpuscular one, thanks to the property of having impulse. Hence the inability to understand why from the infinity of movements that an electron should execute inside an atom according to classical concepts, only some are possible. Enigmas faced by physicists at that time. The need to suppose, for light, two theories, contradictory: undulatory and corpuscular and the inability to understand why from the infinite number of possible motions an electron should execute randomly, according to classical concepts, only some were possible; these were the enigmas that physicists faced at that time. First, the theory of light quanta cannot be considered satisfactory since it defines the energy of the light corpuscle through the relationship W = hν which contains a frequency. But a corpuscular theory does not contain any element that allows the definition of a frequency. For this reason alone, it is necessary, in the case of light, to introduce simultaneously the concept of corpuscle and the concept of periodicity. Anyway, it is necessary to suppose the existence of corpuscles and waves; they constitute, according to Bohm, two complementary facets of reality; it is necessary to be possible to establish a parallel between the movement of a corpuscle and the propagation of the associated wave. Thus we get to the next general idea that led the research for both substance (amorphous matter) and radiation. Particularly for light. It is necessary to introduce simultaneously, the concept of corpuscle and wave. In other words, in all cases it is necessary to assume the existence of corpuscles accompanied by wave. In any case, since the particles and the waves cannot be independent of each other, according to Bohm, they are two complementary facets of the same reality, it must be possible to establish a parallel between the motion of a corpuscle and the propagation of the associated wave. Thus the first objective to be reached was to establish this correspondence. On this occasion we started by considering the simplest case, that of an isolated corpuscle, that is to say, of a 2 corpuscle free of any external influence. Furthermore using the relationship vV = c , between the speed v of the particle and the phase speed V of the associated wave, as well as the corpuscle 2 2 expression (mc ) and undulatory (hv) of the total energy W=m c =hν , de Broglie got to the





particle trajectory, ie where A and B are two points in space through which the ray of light passes, respectively the path of the particle. B

B

∫ d l λ = extremum ←⎯→ ∫ p.dl = extremum

. These concepts led to the interpretation of the stability conditions introduced by quantum theory. Indeed, considering a closed trajectory, it is very natural to assume that the associated wave phase is a uniform function along the trajectory. So we can write: dl ∫ λ = ∫ ( p h )dl = integer A

A

This is exactly the condition of stability of periodic atomic movements. Thus, the condition of quantum stability appears as analogous to the resonance phenomena and the advent of integers becomes here as natural as in string theory and vibrating plates. The content of this volume consists of two parts. The first part is dedicated to the idea that each dynamical system is associated (thanks to variational calculation) with a field theory, which consists of a field, a potential and a current. Their propagation in space is as waves, corresponding respectively to the field, the potential and the current. With these elements, with the help of the theory of harmonic forms, the propagation equations are obtained, as waves, respectively of field, potential and current. The second part highlights the property of the dynamic differential system to provide the space of evolution with a geometry, consisting of a metric and a nonlinear connection. As a result, we obtain a covariant derivation, a parallel transport, self-parallel curves, which are exactly the trajectories of the given system (the solutions of the given differential dynamic system equations), a curvature associated with the evolution space of the given dynamic system, a torsion and selfparallel curves. As a consequence there is a covariant derivative, a parallel transport, self-parallel curves, curvature and torsion.

3

FIELD THEORY

Chap. 0

Page

Theory of harmonic forms

8

§ 0.2

Field Theory

10

§ 0.3

Systems of ordinary differential equations, or with implicit partial derivatives

11

§ 0.1

Part I: Fields associated to Dynamic Systems

17

THE GALLISSOT-SOURIAU FORMALISM   OF CLASSICAL MECHANICS

17

Chap. I § 1.0

History. The Theory of E. Cartan

§ 1.1

Lagrangețs form in classical mechanics (of the material point). A classification 20 of classical dynamic systems

§ 1.2

The direct and inverse problem in g-s formalism

29

§ 1.3

Forms potentially associated to dynamic systems [yy]

33

§ 1.4

The existence and unicity of fields associated to first order dynamic systems

37

§ 1.5

The Gallissot – Souriaut formalism in preferential coordinate systems for 41 dynamic systems written in cynematic form

Chap. II

17

DIFFERENTIAL DYNAMIC SYSTEMS OF THE FIRST ORDER AND FIELDS ASSOCIATED WITH THEM

48

§ 2.0

Integrating factors

§ 2.1

Fields associated to first order (biodynamic) dynamic systems, with an even 51 number of state parameters

§ 2.2

B i o d y n a m i c s y s t e m s w i t h a n o d d n u m b e r m = 2 p + 1 53 of state parameters

Chap III § 3.0

48

GALLISSOT-SOURIAU’S FORMALISM OF CLASSICAL MECHANICS

Observations

56 56

4

§ 3.1

Properties of 2-forms

56

§ 3.2

Almost symplectic and symplectic varieties

69

CHAP. IV

HYBRID DIFFERENTIAL DYNAMIC SYSTEMS

88

§ 4.1

Differential dynamic systems in the general case

88

§ 4.2

Fields that are associated to first order implicit dynamic systems

93

§ 4.3

The special case of second order dynamic systems

102

CAP. V

MULTI-TIME DYNAMIC SYSTEMS ASSOCIATED HYPER-FIELDS

110

§ 5.1

Multi-time first order dds with an even number of state parameters

110

§ 5.2

Multi-time dds, of first order, with an odd number of state parameters

115

§ 5.3

Multi-time dds of second order, associated hyper-fields

117

CHAP. VI

EVOLUTION OF CONTINUOUS DEFORMABLE ENVIRONMENTS ASSOCIATED HYPERFIELDS

127

§ 6.1

Differential continuous systems of first order, with an even number of state parameters

127

§ 6.2

Continuous dynamic systems, of first order, with odd number of state parameteres

131

§ 6.3

Continuous dynamic Systems, of second order, associated Hyerfields

133

Part Two CHAP.VII

GEOMETRIC STRUCTURES ASSOCIATED TO DYNAMIC DIFFERENTIAL SYSTEMS

143

§ 7.0

INTRODUCTION

143

§ 7.1

DDS ASSOCIATED GEOMETRIC STRUCTURES OF FIRST ORDER

143

§ 7.2

GEOMETRIC STRUCTURES ASSOCIATED TO

156

§ 7.3

IMPLICIT DDS GEOMETRIC STRUCTURES ASSOCIATED OF FIRST ORDER,

RHEONOM SYSTEMS

159

SUBJECT TO HOLONOMIC RESTRICTIONS

§ 7.4

GEOMETRIC STRUCTURES ASSOCIATED WITH SUPERFIELDS

166

§ 7.5

GEOMETRIC STRUCTURES, DYNAMIC DIFFERENTIAL SYSTEMS OF THE FIRST ORDER AND THE RELATIONSHIP BETWEEN THEM

176

5



CHAP. VIII

Implicit, Second Order DDS and their geometry

179

§ 8.1

Abstract geometric structures

179

§ 8.2

Geometric structures associated with a second order DDS

184

CHAP. IX

GEOMETRIC OBJECTS AND STRUCTURES ASSOCIATED WITH DYNAMIC SYSTEMS OF DIFFERENTIABLE CONTINUOUS ENVIRONMENTS OF FIRST ORDER

GEOMETRIC STRUCTURES ON JET SPACES

J 1(Rn , M)

190

190

§ 9.1 §9.2

FIRST CUSTOMIZATION: MANIFOLD U IS A TRIVIAL LOCAL FIBRATE SPACE

201

§ 9.3

SECOND CUSTOMIZATION: U MANIFOLD IS A FIBRATE VECTOR SPACE

205

§ 9.4 § 9.5 § 9.6

THIRD CUSTOMIZATION:

U MANIFOLD IS THE TANGENT FIBRATE TM OF 208 A MANIFOLD M

FOURTH CUSTOMIZATION: MANIFOLD U IS THE OSCULATOR 216 FIBRATE Osc2 M THE FIFTH CUSTOMIZATION. CUSTOMIZING THE GEOMETRIC STRUCTURE. SHORT GEOMETRIC STRUCTURE ON TM

221

§ 9.7

THE SIXTH CUSTOMIZATION: CUSTOMIZATION OF THE 223 GEOMETRIC STRUCTURE, ELEMENTARY GEOMETRIC STRUCTURES ON THE Osc2M

§ 9.8

The seventh customization: Customizing The geometric structure, elementary 226 geometric structures on Osc2M

§ 9.9

IMPLICIT SDD OF SECOND ORDER ON M

CHAP. X

§ 10.1

§ 10.2

GEOMETRIC OBJECTS AND STRUCTURESASSOCIATED WITH DYNAMIC SYSTEMS OF CONTINUOUS DIFFERENTIABLE ENVIRONMENTS OF THE FIRST ORDER GEOMETRIC

STRUCTURES ON JET SPACES J 1(Rn ,M)

SYSTEMS OF EQUATIONS WITH PARTIAL DERIVATIVES AND ASSOCIATED GEOMETRIC STRUCTURES

6

229

237

237 248

CHAP. XI

Gallissot-Souriau’s formalism Associated to ordinary dynamic systems of second order.

255

§ 11.1

Lagrange FORM

255

§ 11.2

Potential form

259

§ 11.3

SUPERFIELDS AND SUPERWAVES ASSOCIATED TO NEWTONIAN DYNAMIC SYSTEMS

263

§ 11.4

Dynamic SYSTEMS written in THE MAIN form (quasilinear)

270

§ 11. 5

Superfields ASSOCIATED WITH LAGRANGEAN DYNAMIC SYSTEMS

273

§ 11.6

Gallissot-SOURIAU formalism in preferential references

276

Chap. XII

OPTIMAL DYNAMIC SYSTEMS

282

§ 12.1

General optimization Problems for some dynamic systems

282

Dynamic systems with optimal control

295

Biography

306

§ 12.2

7

CHAP. 0 FIELD THEORY § 0.1 THEORY OF HARMONIC FORMS 1. HODGE'S DE RHAM. H’S ADJUNCTION OPERATOR [17], [55]. Being given a Riemann space V4 and on it a differential α form . With the help of metric g and p 4− p the volume η form, an operator ∗ : Λ (V4 )→ Λ (V4 ) , p = 0, 1, 2, 3 is defined: the form ∗ α is called form α adjoint . This is done by: Definition 1 if

α = α λ1 ,...,λ p dx λ1 ∧ ...dx

∗ α = (∗ α )λ p + 1 ,...,λ4 dx

(∗ α )λ where

(0.1)

(0.1 ')

α

p + 1 ,...,λ4

λ1 ,...,λ p

=

λ p +1

∧ dx

λ p +1

λp

∈ Λ p (V4 )

∧ ... ∧ dx λ4

and p ≠ 0 , then

, form α adjoint has the components defined through:

1 λ ,...,λ η λ1 ,...,λ p ,λ p + 1 ,...,λ4 α 1 p p! ,

= g λ1µ1 ...g

λpµ p

α µ1 ,...,µ p

are α contravariant components . For p=n, ∗ α is a scalar, for p = 0, we define ∗ α = αη

( )

.

The ∗ application is a linear bijection on Λ V4 (it is easy to show that it is injective and surjective). For applications, it is useful to define the linear operator * by its values on the canonical basis of the vector

( ) space Λ V4 , sufficient values to know it in any base and on any p-form. These values are: ∗ 1 = dx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 ,

∗ dx 0 = dx 1 ∧ dx 2 ∧ dx 3 , ∗ dx i = dx 0 ∧ dx j ∧ dx h

(0.2)

∗ dx 0 ∧ dx i = − dx j ∧ dx h , ∗ dx j ∧ dx h = dx 0 ∧ dx i , o

i

j

(i,j,h) ~(0.1,2,3),

h

∗ dx ∧ dx ∧ dx = dx , ∗ dx 1 ∧ dx 2 ∧ dx 3 = dx 0 ,

∗ dx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 = −1. From these formulas it is inferred that: p +1 ∗ 2 = (− 1) id ∧ p ( V ) 4 (0.3) hence: −1 p +1 (0.3 ') ∗ = ( −1 ) ∗ −1 We have, therefore, for the ∗ inverse application values on base: ∗ −1 : 1 = − dx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 ,

∗ −1 : dx 0 = dx 1 ∧ dx 2 ∧ dx 3 ,

∗ −1 : dx i = dx 0 ∧ dx j ∧ dx h ∗ −1 : dx 0 ∧ dx i = dx j ∧ dx h , −1 i j 0 h (0.2 ') ∗ : dx ∧ dx = − dx ∧ dx , (I, J, H) ~ (0.1, 2.3), ∗ −1 : dx o ∧ dx i ∧ dx j = dx h ,

∗ −1 : dx 1 ∧ dx 2 ∧ dx 3 = dx 0 , ∗ −1 : dx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 = 1,

8







OPERATOR δ OF CODIFFERENTIATION

2

p p −1 Definition 2 We call (operator of) codifferentiation, a linear application δ i : Λ (V4 )→ Λ (V4 ) with 2 the property ∀p . δ i = 0 .

Proposition 3 δ1 differentiations

The necessary and sufficient condition as a linear combination of two coδ 1δ 2 + δ 2 δ 1 = 0 . In this situation and δ 2 to be a codifferentiation too is

any other combination of them is also a co-differentiation. A system of k codifferentiations δ i with the property that they anticommute two by two, generates a linear space of codifferentiations, which make use all of the same property of anticommutation. With the help of external differential d, and the operator ∗ , we define the operator δ by the formula: p +1

−1 (0.4) δ = (− 1) ∗ d ∗ p p −1 through which α ∈ Λ (V4 )→ δα ∈ Λ (V4 ) . 2 Proposition 4 The operator δ fulfills the relationship δ i = 0 p p +1 −1 2 −1 −1 2 Indeed, it follows from the definition that δ = (− 1) (− 1) ∗ d ∗ ∗ d ∗ = − ∗ d ∗ null in the base of Poincaré 's lemma: Using in formula (4) formula (3 '), we deduce Properties Being given, in an orthonormal reference where: 0 10 f ∈ Λ (V4 ) f = f (x0, x1, x2, x3 )

20 30

ω =∈ Λ1 (V4 ), ω = a0 dx 0 + a i dx i

Ω ∈ Λ2 (V4 ) Ω = Ai dx 0 ∧ dx i + Bi dx j ∧ dx k ,

Ω ∈ Λ3 (V4 ) Φ = A0 dx 1 ∧ dx 2 ∧ dx 3 + Ai dx 0 ∧ dx j ∧ dx h , 40 Ψ ∈ Λ4 (V4 )Ψ = Adx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 , 50 We have δ f = 0, 10 3 ∂a ∂a ∂a α δω = 00 − ii = , ∂x ∂x α =0 ∂x α 1 ) 0 2 ∂A ⎛ ∂A ⎞ δ Ω = − ii dx 0 + ⎜ 00 + rot B ⎟dx i , ∂x ⎝ ∂x ⎠ (0.5) 30 ∂A ⎞ ⎛ ∂A δ Φ = (rot A)dx 0 ∧ dx i + ⎜ 0i − 0i ⎟dx j ∧ dx h 2 ) ∂x ⎠ ⎝ ∂x 40 ∂A ∂A δ Ψ = i dx 0 ∧ dx j ∧ dx h + 0 dx 1 ∧ dx 2 ∧ dx 3 , ∂x ∂x 50

∑

A form α , for which δα = 0 , is called a coclosed, a form α for which there is a form λ such as δλ = α , is called coexact. Any coexact form is coclosed. The coclosed, respectively the coexact forms constitute vector spaces. 3 LAPLACE – D'ALEMBERT Δ OPERATOR.

Definition 5 Being given a codifferential δ i , it defines a "Laplace" Δi operator by 2

(0.6)

Δ i = (d + δ i ) = dδ i + δ i d : Λ p (V4 ) → Λ p (V4 ) , called Laplacean.

Proposition 6 The application δ i → Δi is linear and injective on any space of anticommutable codifferentiations defined by proposition (4) Definition 0.7 A p-form α, for which Δα = 0 is called harmonic.

9





Any p-form closed and coclosed is harmonic. The set of the p-harmonic forms is, over the body of the reals, a vector space. Proposition 8 The operator Δ commutes with ∗ . p 4 − p +1 −1 −1 −1 (∗ d ∗ d − d ∗ d ∗)= δ d + dδ = Δ , so Indeed, being given ∗ Δ∗ = ∗ dδ ∗ + ∗ δd ∗ = (− 1) (− 1) (0.7) Δ ∗ − ∗ Δ = 0 . Properties of Δ operator

0.10

∀f ∈ (Λ4 ) , we have Δ

f =

∂2 f 0 2

(∂x )

− Δ0 f

3

To demonstrate property 20 is sufficient to check the relationship for the forms ω , Ω, Φ, and Ψ . 0.20

∀α = α λ1 ,...,λ p dx λ1 ∧ ... ∧ dx 3

(0.8)

Δf = ∑ 1

λp

∈ Λ p (V4 )

, we have:

(

)

Δα = Δ α λ1 ,...,λ p dx λ1 ∧ ... ∧ dx

λp

2

∂ f (∂x i ) 2 .

0.2 FIELD THEORY The classic theory of defining and studying in general well-known differential operators serves to study the electromagnetic field. We will/generalize these notions in this paragraph. Being given a space Riemann M and on it a 2- a closed form (da= 0); we will associate to it as follows, a field theory: 1 ° Being given a closed a 2-form, it is associated as known, locally (whereas 2-closed form is exact locally) a 1-form l , so a = dl; 1-form l is called potential form. Being given a potential form l, the form of the field a is uniquely determined; reciprocally, given form a , its potential is determined ignoring a total exact differential. Any other 1-form λ = λ + dϕ is also a potential for the field α . 2 ° Using operator ∗ Hodge de Rham’s of adjunction, alongside 2-form a is highlighted too a (m-2)-form b = ∗ a. Together, forms a and b define what we will call a field. To define the field is sufficient to know one of these two forms, each of them leading to the other through ∗ , as we will see below. 3 ° with 1-form l is associated (m-1)-form x by formula l = ∗ x. Together, forms l and x define for the given field what we will call potential. −1 −1 4o Applying ∗ to relationship λ = λ + dϕ we obtain ξ = ξ + ∗ dϕ , we choose the function ϕ n+1 so dξ = 0 . It is a solution of Poisson Δ( ∗ϕ ) + ( −`) dϕ equation , 5 ° In general the b form is not closed, so there is a relationship as Db+g= 0, where g is a (m-1)form. The form g is called current form together with 1-form q = ∗ g. 6 ° From the definitions given above, the following Maxwell type equations are deducted. For potential: dx= 0, dl-a= 0; for field da=0, db+g=0; for current dg= 0, d q -n= 0. 7 ° Using the operator of Laplace-d'Alembert D , the following equations called of propagation are obtained . For potential Dx=ddx, for field Da=dda, respectively for current Dg=ddg. 8o Equation dg= 0 is known as the continuity equation. 9 o An important equation, which defines potential through the current, is the equation of propagation of the potential: Dl= ∗ g Schematically, these results can be presented in the following table:

10

Potential

Field

Current

Forms of definition:

xÎLm-1(M), l=*xÎL1(M),

aÎL2(M), b=*aÎLm-2

gÎLm-1(M), q=*gÎL1(M),

Maxwell type equations (field):

Dx=0, l-d=0 ,a

Da= 0, Db+g= 0,

Dg= 0, Dqn-=0,

Propagation equations (of waves):

Dx=ddx,

Da=dda,

Dg=ddg.

Equations of Maxwell type (field)

dl=0, dx+b=0,

db=0, da-q=0,

dq=0, dg+n=0,

Propagation equations (of waves)

Dl=ddl,

Db=ddb,

Dq=dd

As we will see below, each dynamic system (spray) is associated with a W closed 2-form. It acts as 2-form a and therefore is associated with a field theory and, consequently, a field and waves, as solutions of the propagation equations. These results will be adapted by generalization to the case of the TM manifold We'll see in part three that dynamic systems, first or second orders, are associated to Lagrange forms and, accordingly, fields and where. Then we will develop this idea and we'll show its generality.

§ 0.3 SYSTEMS OF ORDINARYY DIFFERENTIAL EQUATIONS, OR WITH IMPLICIT PARTIAL DERIVATIVES To study the evolution of a dynamic system, we believe that it is modeled by a system of ordinary differential equations, with partial derivatives respectively, written under implicit form. Why implicit? Currently, we are not working with equations written under implicit form. The implicit form of the equations of evolution is obviously more general (there are implicit equations that cannot be explicit), but this is not the real reason why we adopted this view. The study of equations as implicit, globally and correctly defined, is simpler and more geometrizable, following then the study of explicit ones to be regarded as a particular case of the former. Largely, the study of ordinary differential equations can be extended to partial derivatives equations. A geometric framework for the presentation of dynamical systems theory of continuous environments requires consideration of differentiable manifolds equipped with certain structures. Differentiable manifolds considered are real, finite dimensional, Hausdorf and paracompact. Different applications (functions) that occur are supposed to be class C The notion of a fibrate means local trivial fibrate p:E®M (noted, for simplicity, with E as well) with base space M, a differentiable manifold of dimension m and the total space is a variety of dimension m + n. The local coordinates of the fibrate E will be denoted by (xi, u) with i= 1, n and a= 1, m , and local sections ∞

set on an open group of M, by SectE/U={s:M®E½p*s(x)=x, "xÎU}. We denote by T(E) and T(M) the beam of tangent vectors respectively E and M. 0.3.1 Definition of systems of implicit differential equations, ordinary, covariant, of order p. Given Fn a section of

δ n1 : *

(0.3.1) Fn : (t, x, x ,...,x(n))ÎJnM®F(t, x, x ,...,x(n))ÎT x M, which highlights the set: Ker Fn = {(t, x, x ,...,x(n))ÎJnM½Fn(t, x, x ,...,x(n))=0}; n we say that Ker F defines a differential equation of order n, implicit. In local coordinates (in map ( U ,Φ )) function Fn is represented by:

 

 

 

 

 

 

 

 

 

 

11

 

 





Nature

 

n

h

h

( n )h

 ) dxi, "tÎI, (t, x, x ,...,x(n))ÎJnM, (0.3.2) Fn = F i ( t , x , x ,..., x and the condition Fn = 0 is expressed by the relationships: n h ( n )h h ) =0, (i,h= 1, m ), (0.3.3) F i ( t , x , x ,..., x named system of differential equations of order n implicit, ordinary (nondegenerate) if they satisfy the ∂F ∂F ( ( ni) j )x ( ( ni) j ) relationship det ∂x ¹ 0, for any xÎU (condition that, in future, will be understood writing det ∂x ¹ 0),), and singular otherwise. The set KerFn (JnM's differentiable submanifold of codimension m) leads to the total space Ker Fn ´MJoM of the fibrate space F = (Ker F´MJnM,pn,JnM), subfibrate of tnM = (JnM´MT*M,pt,JnM). i i h n h ( n )h h ) ,, the On a local map change on M, through which x = x ( x ) , functions F i ( t , x , x ,..., x n

local components of F i change by the formula: ∂x h Fi n = i Fhn ∂x (0.3.4) and thus constitute components of a covector distinguished on manifold M. Solutions of differential equations of order p Given: c:tÎIÌR®x=c(t)ÎUÌM, a differentiable curve. This is lifted to JnM by: dc( t ) d n c( t ) c : t ∈ I → ( t , c( t ), ,..., )∈ J n M . dt dt n We consider the mutual image of F by c as function:

dc( t ) d n c( t ) ,..., ]. dt dt n Definition 0.3.1 We call local solution of the system of differential equations (0.3.3), a function c: tÎIÌR® x=c(t)ÎUÌM, which makes use of the property: c ∗ F n = F n .c = F n [ t , c( t ),

dc( t ) d n c( t ) ,..., ] ≡ 0 , ∀t ∈ I . dt dt n In the case of ordinary differential equations, the implicit differential equations system (0.3.3) is equivalent (has the same solutions) with the system written in kinematic form: (0.3.5) x(n)i = fi(t, x, x ,...,x(n-1)), (i= 1, m ), obtained by expressing the variables x(n)i Fi n [ t , c( t ),

n

The system (0.3.5) defines a section f in the vector fibrate (JnM,p n−1 ,Jn-1M), where n

p n −1 :(t, x, x ,...,x(n))®(t, x, x ,...,x(n-1)). In the set of ordinary differential equation systems, implicit, covariant, of order p, locally written as:

(

)

Fi t , q , q ,..., q ( p ) = 0 we can define a relationship of equivalence. D ij t , q, q ,..., q ( p )

(

Let’s consider this a mixed tensor D of local components

( )

)with the property that

i ∀(t , q )∈ J p M , det D j ≠ 0 (in any open case).

All mixed tensors, above defined, is organized as a group in relation to the law of composition:

~ ~ ~ Di j = Dih Dhj

having as neutral element δ of Kronecker.

By definition we say that two systems of equations of components conditions

Gi = Di j F j

(Fi ) and (Gi ) are equivalent if the

are equivalent.

(F )

(G )

i By definition, it follows that whatever system solution, i it is also a solution for the system (is solution for any equivalence class systems defined by Fi). 0.3.2 Definition of systems of ordinary differential equations, implicit, contravariant of order p

12

 

KerG =

)

{

(

G t , q , q ,..., q ( P )

(

)

} which in local writing, is expressed )= G (t ,q ,q ,...,q ) ∂ ∂q i

(P)

i

. p (KerG set (submanifold of JpM too) provide total space of fibrate space F=( KerG × M J M ,π, JpM),

J p M × TM , p ,

M 0 fp,JpM).) subfibrate of space ( On a change of map, the local components of G are changing by the formula:

Gh =

∂q h i G ∂q i

Any differentiable curve c : t ∈ I ⊂ R → c( t ) ∈ U ⊂ M is lifted by:

⎛ dc( t ) d p(t )⎞ ⎟ ⊂ J pM c : t ∈ I → ⎜⎜ t , c( t ), ,..., p ⎟ dt dt ⎝ ⎠ . Definition 0.3.2 We call local solution of the system of equations (0.3.3) a function c that makes use of the property that:

⎛ dc( t ) d p(t )⎞ ⎟ ≡ 0 , ∀t ∈ I G i ⎜⎜ t , c( t ), ,..., dt dt p ⎟⎠ ⎝ . And the set of contravariant systems of differential equations we define (by the same group {D}) an equivalence relation:

~ G i = Dhi G h

Between the set of equivalence classes of covariant dynamical systems and the contravariant ones, a bijection can be established. Given D a second order tensor twice covariant (distinguished) nondegenerate, of components Dij (det (Dij) ≠ 0 and being given its inverse (Dij). Then any system of covariant differential equations Fi = 0 is associated to the contravariant system

~ F i = D ij F j

and vice versa.

{~ }

{F }

i

Covariant differential systems i of a class and the corresponding class contravariant F accept the same solutions. We bring together these systems and will call their reunion a representative of which, D, is given the ” dynamic” of that system. The goal we seek to meet in this paragraph, is to study the dynamic Newton systems of second order). We emphasize self-adjunction conditions for systems written in main form, respectively in kinematic form. 0.3.3 Implicit covariant ordinary differential equations linearization

Being given the system of equations

Fi ( t , q , q ,..., q ( p ) ) = 0 ,

det( with the property

it we associate the development in Mac-Laurin formula, compared to (

Fi ( t , q , q ,..., q

(P)

0

) = Fi ( t , q0 , q 0 ,..., q

(P) 0

Fi ( t , q , q ,..., q

⎛ ∂Fi ⎜⎜ ( p ) j ⎝ ∂q where Aij=



 

 

(

G : t , q , q ,..., q ( P ) ∈ J p M → G t , q , q ,..., q ( P ) ∈ Tq M ∀q ∈ M , and its core p (t , q , q ,..., q )∈ J M G (t , q , q ,..., q ) = 0

Given a function,



 

Similarly, we can define implicit ordinary differential equation systems, contravariant.

(

∂Fi

) + ∂q

(P)

( p)j

) = Aij q

q

( p )i 0

∂Fi ∂q ( P )

j

) ≠ 0, to

)

j

j

)o ( q ( p ) − q ( p ) 0 ) + R2 ( t , q , q ,..., q ( p ) )

( p)j

,

+ Bi + R2

⎞ ⎟⎟ j ⎠ qoi and B = Fi 0 ( t , q0 , q 0 ,..., q0( P ) ) - Aij q0( P ) . i 13

=0,

 

 

Definition 0.3.3 We say that a dynamic Newtonian system of the form (1.2.1) is written as main form, if its equations are of the form: (1.2.6)

Fk = Aki(t, q,..., q

( p −1 )

)q

( p − 1 )i

+Bk(t, q,..., q

( p −1 )

)=0,

∞

with Aki, BkÎ C (J1M), det(Aij)¹0. j

Aij q ( p ) + Bi

The dynamic system = 0, approximates the given system. It is written in the main form and, by definition, it is proper. A special case of dynamical systems written in the main form is the kinematic form of the systems: i

q( p ) = X i , system that is contravariant.

i

i

Dynamic first order systems q = X , another presentation form of equations which are deducted, lead to the kinematic form that is symmetrical:

dq 1 dq n dt = ... = = 1 h n h X ( t ,q ) 1 . X ( t ,q )

For development, see Chapter. XIII.

0.3.4 Implicit systems of ordinary differential equations of the first order, covariant Given a function F:J1M®T*M, which can be regarded as a section in fibrate F:(J1(M)´MT*M, p1, J1M). Locally, on an open set U, local map geometric area, and on a closed interval I of the real axis we have: *     F:(t, q, q )®F(t, q, q ) = Fi(t, q, q )(dqi)qÎT x M , (tÎI, (t, q, q )ÎJ1(U)).

⎛ ∂Fi ⎜ ⎜ ∂q j We make the assumption det ⎝

⎞ ⎟ ⎟ ⎠ ¹ 0 and we put out the set:   KerF = {(t, q, q ) ½ F (t, q, q ) = 0}; i

defining first order system in local writing:

 (0.3.7) Fi(t, q, q ) = 0, system named system of the first order of implicit ordinary differential equations. Fi =

∂q h Fh ∂q i and thus

i i h On a change of local map q = q ( q ), functions Fi change by the formula: constitute components of a distinguished covector. ⎛ ∂Fi ⎞ ⎜ ⎟ ⎜ ∂q j ⎟ ⎠ ¹ 0, made above, the system (3.0.1) is called ordinary or regular, and singular Assuming det ⎝ otherwise. Any differentiable path c: tÎIÌR ® q = c(t)ÎU is lifted J1(M), by dc( t ) ⎞ ⎛ ⎜ t , c( t ), ⎟ dt ⎠ ÎJ1(M). : c : tÎIÌR ® ⎝

Let’s consider function’s F by c , mutual image, namely the function: dc( t ) ⎞ ⎛ ⎜ t , c( t ), ⎟ dt ⎠ ÎR. tÎIÌR ® c *F = (F. c ) = F ⎝ A (local) solution of the system of differential equations (3.0.1) is a function c:tÎI ® q = c(t)ÎU, which makes use of the property: dc( t ) ⎞ ⎛ ⎜ t , c( t ), ⎟ dt ⎠ º 0, " tÎI. Fi ⎝ The set of solutions to equation (3.0.1) forms locally (is construed as) a congruence of curves on J10(M) = TM 0.3.5 Implicit systems of ordinary differential equations of second order, covariant

14

with coordinates (qi, vi). The differential equations of trajectories of the field F on R × TM are: dq i dv i dt = vi , = F i ( t , q , v ), = 1, dt dt dt (0.3.8) and second order differential equations of projections of above trajectories on M are: d 2qi dq = F i ( t ,q, ) 2 dt dt (0.3.9) equations that model the dynamics of a non-autonomous system. Given now a change of local maps now on manifold M defined by formulas (0.3.10) x i = x i ( x 1 , ..., x m ) . This induces a change on vector maps appropriate on TM by

q i = q i (q h ), v i =

∂q i j v . ∂q j

(0.3.11) Compared with vector maps, field components S change by formulas

vi =

∂q i j ∂v i j ∂v i i v , F = v + j F ∂q j ∂q j ∂v

(0.3.12) The differentials of the functions (0.3.11) are:

dq i =

∂q i h ∂v i ∂v i h i h dq , d v = dq + dv . ∂q h ∂q h ∂v h

(0.3.13) 0.3.6 Systems of equations with implicit partial derivatives of first order, covariant Given differentiable manifolds M = Mm and N =Nn respectively m and n of finite size and local coordinates (xi) and (ua) and Jo(N,M) = N´ M their Cartesian product. Manifold N is called space of parameters and M, configuration space, which we will assume for the moment, of even size m = 2p . The product manifold is regarded as a trivial fibrate space, evolution space or phase space. We note with J1(N,M) a

∗

´MT*M the fibrate space with base J1(N,M) and with fiber at the point (xi,ua,u i ): space of covectors T x M. A system of partial derivatives equations, implicit, of first order, described by a definition similar to the ∗ second order systems, is defined as describing the core of an application F : ( N , M ) → 0 ∈ T M .. It is so locally written as:

i b b (0.3.14) Fa( x , u , u i ) = 0. We say that the system (0.3.4.1) is proper if it has the property: ⎛ ∂F ⎞ det ⎜⎜ ab ⎟⎟ ≠ 0 , ( ∀i ∈ 1, n ) ⎝ ∂u i ⎠ (0.3.15). By definition we have on a change of the local map on M , of the form , functions Fa change by the formulas: ∂u b Fa = a Fb ∂u Let’s consider a local map change on M, it induces c preferential map change on J1(N,M), as: x i = x i ( x h ), u a = u a ( x i , u b ), ∂x h ⎛ ∂u a ∂u a ⎞ u ia = i ⎜⎜ b u hb + h ⎟⎟. ∂x ⎝ ∂u ∂x ⎠

 

15



 

Now being given the differentiable manifold TM, where M is a differentiable manifold of dimension m. Let’s consider on the space of evolution R × TM a base of contact forms whose local expression isqi = dqi - vidt, where (qi, vi) is an arbitrary system of coordinates on the manifold TM. Given (U , ϕ ) a local map on M with coordinates (qi), respectively (U , Φ ) appropriate vector map on TM

a

Among the systems of partial derivatives equations a special role have linear equations, namely the pseudo-

∂f X ( f ) ≡ X y i = 0 respectively X ( f ) ≡ X y i = R where y = ∂x i linear ones. They are of the form: i i

i

0.3.7 Systems of partial derivatives equations, implicit, of second order

1 The definition of partial derivatives equation systems, covariant of the second order, implicit Being the differentiable manifolds M and N, being associate the fibrate space: 2 ∗ 2 . J ( N ,M ) ×M T M , pM , J ( N ,M )

(

)

We note with F a section in this space, that is a function of the form F: (

and

x i , u a , u ia , u ija

{(

)ÎJ2(N,M)®F(

x i ,u a ,u ia ,u ija ∈ J 2 ( N , M ) × M T ∗ M ) ,

)

) }

(

KerF = x i , u a , u ia , u ija ∈ J 2 ( N , M ) F x i , u a , u ia , u ija = 0

its core.

x i , u a , u ia , u ija Definition 0.3.4 Condition F ( ) = 0 is called equation with partial derivatives of second order, implicit.. In local coordinates, we have: x i , u b , u ib , u ijb (0.3. 16) Fa( ) = 0. We say that these functions are called system of partial derivatives equations of second order, implicit. a a b On a change of local map on M, u = u ( u ) , functions F change by the formulas: a

b

∂u Fb ∂u a (0.3.17) and constitute components of a distinguished covector (d-covector). We say that the system (0.3.8) is proper if: ⎛ ∂F ⎞ det ⎜ ab ⎟ ≠ 0 , ( ∀i , j ∈ 1, n ) ⎜ ∂u ij ⎟ ⎝ ⎠ (0.3.18). 2 Solutions It is called solution of equation (0.3.15) a system of functions ua =ua(xi), , so that ⎛ ∂u b ∂ 2 u b ⎞ Fa ⎜⎜ x h , u b ( x h ), h , h k ⎟⎟ ≡ 0 , ∀( x h ) ∈ N ∂x ∂x ∂x ⎠ ⎝ (0.3. 19). The study of dynamical systems, ordinary, differential, of the first order [ ] or second order[ ] revealed an interesting property of theirs, namely to be associated fields and waves propagation of the respective fields, in certain well defined corresponding spaces. Generalizing these considerations, in the case of evolution of continuous deformable environments, with applications in biology, for example, is promising for the purposes stated above. Fa =

 

 

16



 

Being given a system of equations of the form (0.3.14). It's called solution a multitude of functions ∂u c = u hc ( x k ) h ua = ua(xh), so that included with their partial derivatives ∂x in equations (0.3.14) we obtain the identities: i b h b h h F ( x , u ( x ),u i ( x ) ) ≡ 0 , ∀( x ) .

PART ONE: FIELDS ASSOCIATED TO DYNAMIC SYSTEMS CHAPTER I. THE GALLISSOT-SOURIAU FORMALISM OF CLASSICAL MECHANICS This chapter briefly presents the history of the Gallissot - Souriau (G-S) formalism, adapted to the development of the idea of corpuscular - wave unity.

§ 1.0 HISTORY. THE THEORY OF E. CARTAN The history of the birth and development of the idea of associating the dynamics of material systems with what we nowadays refer to as a ”field” has, as a starting point, the character of the photon to manifest itself in some experiments as having a corpuscular nature (impulse particle), in others having an undulating nature. This led the illustrious scholar Louis de Broglie (1924) [2] to admit the simultaneous existence of these properties (wave optics). The study of field properties, as well as those of the waves, can be done by means of geometrization, by using external differential forms. The use of external differential forms in mechanics in general and particularly in dynamics, in fact, has its starting point in Lagrange's work [11]. In the current language, however, one can speak of it with Gallissot’s works [6], [7] and [8], works that outline the first stage of the birth of the formalism that we present in this chapter. In this context, J. Klein's works [9], [10] are also to be quoted. In the description of material point movement, Newtonian classical equations (1.0.1), written for a material point, were replaced by Gallissot, in a Note that is considered fundamental for this problem [6], by giving a differential 2-form, called by Souriau Lagrange’form, [39] which we present in the following paragraph. The geometrization of this problem can be considered as having the starting point in Gallissot's works, cited above, which capture the dynamics of material systems in Newtonian mechanics first but also in restricted relativity by considering a differential 2-form. The second stage of development is the one that postulates the existence of such a (closed) 2-form. It was formulated and placed at the basis of classical mechanics in his famous book "Géométrie et relativité" by J.-M. Souriau [39]. With these regrouped ideas, it is now easy to mathematically prove - through the formalism, called Gallisot-Souriau, that the evolution of any dynamic system can be viewed either as a corpuscular nature, through (Newton type) second order differential equations, ordinary formalism, or as a wave nature, of the existence of a (super) field and its propagation waves, respectively, in an appropriately constructed evolution space. Regarding the ordinary dynamic character, the existence of such a 2-form, whose characteristics coincide with the trajectories of the system, is ensured by one of Cartan's theorems. The 2-form constructed by Gallissot is canonically associated to the system. There is an equivalence class of such 2-forms, among which a closed one can be chosen, whose uniqueness is ensured under the given initial conditions. 1 Classic dynamic systems and associated fields In this chapter it is shown how any non-autonomous semi-spray (a non-autonomous second order differential equation system) is canonically associated with a field. Lagrange's form is defined and Maxwell's principle is formulated. The history is presented, as well as the basis for the construction of a field theory with which we will deal later, in the following chapters. The logic of the association of this theory is highlighted, and the bases are set for the Gallissot Souriau formalism as a method of study. The components of the field are the coefficients of the W Lagrange 2-form, closed and of maximum rank (Maxwell's principle).  In the Newtonian formalism of classical mechanics, for a mechanical system, a force field F=F(t,q, q ) i (dynamic system) is given and, based on Newton's equations: q =Fi, (m=1) (more exactly, Fi represents the acceleration field), which is the second principle of mechanics, the system dynamics is determined: qi=qi(t,c1,...,c2m).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

​​

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

17

 

 



 

Science means simply the aggregate of all the recipes that are always successful. All the rest is literature. Paul Valery

Let M be a differentiable variety of C¥ class and of finite dimension m, on which a dynamic system S is i defined. Its evolution is described by a system of Newtonian differential equations of the form q =Fi, where the force field Fi is assumed to be differentiable. Newton's evolution equations expressed in the evolution space (space of states) J1(M)=R´TM can be written as a system of first order differential equations in the Pfaffian form:

dv i − F i dt = 0 , dq i − v i dt = 0.

(1.0.1)

The evolution of such a system is defined by giving the force field F=F(t,q,v), called the dynamic system. The trajectories of the material points (of mass 1) placed in the force field F, can be obtained as solutions of system (1.0.1) determined by some initial conditions; they describe the state of evolution of the system. One of the problems studied in mechanics is to determine the positioning functions qi and, with i them, speeds vi = q , as time functions, under the given initial conditions, with which the dynamics of the respective system is known.

1.0 HARMONIC FORM THEORY 1. HODGE – DE RHAM.H [17], [55] ADJUNCTION OPERATOR Let V4 be a Riemann space and α a differential form in it. With the aid of the metric g and volume p 4− p form η , an operator is defined as ∗ : Λ (V4 )→ Λ (V4 ), p=0,1,2,3: form ∗ α is called the adjunct of form α . This is done through: Definition 1 If

α = α λ1 ,...,λ p dx λ1 ∧ ...dx

∗ α = (∗ α )λ p + 1 ,...,λ4 dx

λ p +1

∧ dx

(∗ α )λ (1)

λ p +1

p + 1 ,...,λ4

(1’)

∈ Λ p (V4 )

∧ ... ∧ dx λ4

=

where:

α

λp

λ1 ,...,λ p

and p ≠ 0 , then

, the adjunct of form α has the components defined by:

1 λ ,...,λ η λ ,...,λ ,λ ,...,λ α 1 p p! 1 p p + 1 4 ,

= g λ1µ1 ...g

λpµ p

α µ1 ,...,µ p

are the counter-variant components of α . For p=n, ∗ α is a scalar, for p=0, ∗ α = αη is defined.

( )

Application ∗ is a linear bijection on Λ V4 (it is easy to show it is injective and surjective). With a view to the applications, it is useful to define the linear operator ∗ through its values on the canonical

( )

basis of vector space Λ V4 , sufficient values to know it in any base and any p-form. These values are:

∗ 1 = dx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 ,

∗ dx 0 = dx 1 ∧ dx 2 ∧ dx 3 , ∗ dx i = dx 0 ∧ dx j ∧ dx h

(2)

∗ dx 0 ∧ dx i = − dx j ∧ dx h , ∗ dx j ∧ dx h = dx 0 ∧ dx i , o

i

j

h

∗ dx ∧ dx ∧ dx = dx , ∗ dx 1 ∧ dx 2 ∧ dx 3 = dx 0 ,

∗ dx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 = −1. Through these formulas, it is deduced that: p +1 ∗ 2 = (− 1) id ∧ p ( V ) 4 (3) resulting: ∗ −1 = ( −1 ) p +1 ∗ (3’)

Thus, for the inverse application ∗

−1

we have the base values:

18

(i,j,h) ~(1,2,3),

∗ −1 : 1 = − dx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 , ∗ −1 : dx 0 = dx 1 ∧ dx 2 ∧ dx 3 ,

∗ −1 : dx i = dx 0 ∧ dx j ∧ dx h ∗ −1 : dx 0 ∧ dx i = dx j ∧ dx h , (2’)

∗ −1 : dx i ∧ dx j = − dx 0 ∧ dx h , −1

o

i

j

(i,j,h) ~(1,2,3),

h

∗ : dx ∧ dx ∧ dx = dx ,

∗ −1 : dx 1 ∧ dx 2 ∧ dx 3 = dx 0 , ∗ −1 : dx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 = 1, 3 CO-DIFFERENTIATION OPERATOR δ p p −1 Definition 2 We will call a co-differentiation (operator) a linear application δ i : Λ (V4 )→ Λ (V4 ), 2 with the property that ∀p . δ i = 0 .

Proposition 3 The necessary and sufficient condition for a linear combination of two coδ 1δ 2 + δ 2 δ 1 = 0 . In this situation, any differentiations δ 1 and δ 2 to also be a co-differentiation is for other combination of these is also a co-differentiation. A system of k co-differentiations δ i , with the property

that two by two anti-commute, generates a linear co-differentiation space, all of them with the same anticommuting property. With the aid of the exterior differential d, and the operator ∗ , we shall define the operator δ through the formula: p +1 δ = (− 1) ∗ −1 d ∗ (4) p p −1 Through which α ∈ Λ (V4 )→ δα ∈ Λ (V4 ). 2 Proposition 4 The operator δ verifies the relation δ i = 0 p p +1 −1 2 −1 −1 2 Indeed, from the definition it results that δ = (− 1) (− 1) ∗ d ∗ ∗ d ∗ = − ∗ d ∗ , null based on the Poincaré lemma: By using formula (3’) in formula (4), we deduce that Properties Let an orthonormal coordinate system be in which 0 10 f ∈ Λ (V4 ) f=f(x0, x1,x2,x3 )

We have:

1 20 ω =∈ Λ (V4 ), Ω ∈ Λ2 (V4 ) 30 Ω ∈ Λ3 (V4 ) 40 4 50 Ψ ∈ Λ (V4 )

ω = a0 dx 0 + a i dx i Ω = Ai dx 0 ∧ dx i + Bi dx j ∧ dx k ,

Φ = A0 dx 1 ∧ dx 2 ∧ dx 3 + Ai dx 0 ∧ dx j ∧ dx h ,

Ψ = Adx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 ,

10 δ f = 0 , 3 ∂a ∂a ∂a α δω = 00 − ii = , ∂x ∂x α =0 ∂x α 1 ) 20 ∂A ⎛ ∂A ⎞ δ Ω = − ii dx 0 + ⎜ 00 + rot B ⎟dx i , ∂x ⎝ ∂x ⎠ 30 ∂A ⎞ ⎛ ∂A δ Φ = (rot A)dx 0 ∧ dx i + ⎜ 0i − 0i ⎟dx j ∧ dx h 2 ) ∂x ⎠ ⎝ ∂x 40 ∂A ∂A δ Ψ = i dx 0 ∧ dx j ∧ dx h + 0 dx 1 ∧ dx 2 ∧ dx 3 , ∂x ∂x 50

∑

(5)

19







A form α , for which δα = 0 , is called co-closed, a form α for which there is a form λ so that δλ = α , is called co-exact. Any co-exact form is co-closed. The co-closed, respectively co-exact forms, constitute vector space. 4 THE LAPLACE – d’ALEMBERT Δ OPERATOR Definition 5 Being given a co-differential δ i , it defines a „Laplace” operator Δi by 2

Δ i = (d + δ i ) = dδ i + δ i d Λ p (V4 ) → Λ p (V4 ) : , called Laplacian.

(6)

Proposition 6 The application δ i → Δi is linear and injective on any anti-commutable codifferentiation space defined through proposition (4) Definition 7 A p-form α, for which Δα = 0, is called harmonic Any closed and co-closed p-form is harmonic. The array of harmonic p-forms constitutes, over the real body, a vector space. Proposition 8 Operator Δ commutes with ∗ . p 4 − p +1 −1 −1 −1 (∗ d ∗ d − d ∗ d ∗)= δ d + dδ = Δ be, therefore Indeed, let ∗ Δ∗ = ∗ dδ ∗ + ∗ δd ∗ = (− 1) (− 1) Δ∗−∗Δ =0 .

(7) Properties of the Δ operator

∀f ∈ (Λ4 ), we have Δ

10

f =

∂2 f 0 2

(∂x )

− Δ0 f

3

To prove the 20 property, it is sufficient to verify the relation for forms ω , Ω, Φ, and Ψ . 20

∀α = α λ1 ,...,λ p dx λ1 ∧ ... ∧ dx

λp

(

)

∈ Λ p (V4 ) Δα = Δ α λ1 ,...,λ p dx λ1 ∧ ... ∧ dx , we have:

λp

∂2 f Δf = ∑ i 2 1 (∂x ) . 3

(8) § 1.1 LAGRANGE’S FORM IN CLASSICAL MECHANICS (OF THE MATERIAL POINT). A CLASSIFICATION OF CLASSICAL DYNAMIC SYSTEMS 1.1.1 ‘SLAGRANGE FORM. THE GALLISSOT FORMALISM The Lagrange form, which is presented below, has as its purpose the geometrization of evolutionary equations of dynamic systems. This collection of results outlines the manner of building a unitary theory of the link between the evolution of dynamic systems, regarded in an isolated way, and the hypothetical existence of some fields associated with these systems. 1 Lagrange's simple form In the following lines we will consider the particular case M = R 3. Let M be a differentiable variety of size m=3. In the Newtonian formalism of classical mechanics, the study of the evolution of a system of material points is done by considering Newton's second principle, expressed in the space of evolution (space of states) R×TM, by the differential equations: (1)

m(i) dv i − F i dt = 0 , dq i − v i dt = 0 , i = 1,m.

(2)

Ω3 = mdv i − F i dt ∧ dq i − v i dt = mdv i ∧ dq i + F i dq i − mv i dv i ∧ dt

The accomplishment of this goal of geometrization consists, in essence, in terms of evolutionary equations, of the well-known theorems of Elie Cartan (Theorem 1.0.1); based on this theorem, a 2-form defined on R×TM, of class 2m, is associated as a characteristic field of direction, the field defining the evolution of a mechanical system described by Newton's equations associated with it. In describing the evolution of the material point, Newtonian classical equations (1.1.1), written for a material point, were replaced by Gallissot, in a Note considered fundamental to this problem [7], by giving a differential 2-form, called by Souriau [39], Lagrange's form.

(

) (

)

(

(summing is done after i = 1, 3 ). This can be done on the basis of the following theorem:

20

)

Theorem 1.1.1 To any dynamical system describing the evolution of a material point of mass m, of coordinates qi, animated by a vi component velocity, subjected to a force F of Fi components, in relation to an orthonormal Galilean trihedral, there can be associated a 2-form (1.1.2) possessing the following properties: 1) Form Ω3 is invariant to transformations of the Galilean group (inertial reference change group):

q i = a ij q j + a i t + b ,i t = t + t0 , v i = a ij v j + a i ,

(1.1.3 ) i

(a ) where j Î SO(3), i, j = 1,3 ; in other terms, the form Ω3 has the same expression in relation to any orthonormal Galilean coordinate system. 2) Differential equations of the point evolution are associated with the 2-form Ω3 in Cartan’s sense (the trajectories of motion being the characteristics of Ω3). 3) Ω3 is unique. 4) Ω3 is expressed uniquely according to the differentials of the prime integrals of the motion equations, its coefficients being the components of an antisymmetric tensor, depending on the prime integrals and a variable t. If dΩ3 = 0, then Ω3 is expressed only with the help of the prime integrals and their differentials. This method allows on the one hand formulating results in a more symmetrical form, and on the other hand using in the study of mechanics a more powerful mathematical apparatus - Cartan's external calculation. Form Ω3, given by (1.1.2), highlights in its composition three parts: -a symplectic part: w = mdviÙ dqi, -a kinetic part (reduced): mdviÙdt, both closed, and -a dynamic part Fi dqiÙ dt (also reduced). The kinetic part implies the existence of the exact form dT = mvidvi, where T = mv2/2, a function defined on T(R3) and called kinetic energy. The dynamic part highlights the semi-base 1-form a = Fidqi, called elementary mechanical work. The form (1.1.2), invariant in relation to the Galilean group (1.1.3) and of rank 6 is associated with a characteristic directional field, whose tangent curves are exactly the graphs of the system solutions (1.1.1). Indeed, the characteristics of the 2-form Ω3 are described, as is known, by the equations:

i ∂ Ω 3 = 0, i

i ∂ Ω3 = 0 i

∂q (1.1.4) ∂v , and that are formally obtained from Ω3 by:

∂Ω 3 =0 i ∂ ( dv ) (1.1.4 ') ,

∂Ω 3 =0 ∂( dq i ) .

∂Ω 3 =0 The relation ∂( dt ) is a consequence of the first and represents the kinetic energy theorem. In the following lines we will call Ω3 the simple Lagrange form. The geometric structure of the space of evolution is related to Galileo's group, associated with inertial and reference changes. 2 Lagrange’s expanded form [39] Mechanical reasons force us to take into account, apart from the inertial coordinate systems, rotation coordinate systems in relation to the first. Such a coordinate system change is given by formulas as follows:

q = A( q + r ), t = t + t ,

0 (1.1.5) where A and r are differentiable functions of t. The transformation group (1.1.5) is called the Coriolis group. From the fact that the matrix A(t) Î SO(3), the identities are deduced: At A = I

(1.1.6)

A( x ).A( y ) = x. y A( x ) × A( y ) = A( x × y ) . The last formula is transcribed as:

21































−1 (1.1.7) j(A(x)) = Aj( x ) A , by using the vector product operator j(x) : y ® x × y . From the above relations it follows that there is a vector R Î T(R3) dependent on t, defined by: dA = − Aj ( R ) . dt (1.1.8) This vector is called the instantaneous rotation vector of the new coordinate system in relation with the initial coordinate system, its components being given by: da kl R i = a lj , dt where (i,j,k) is an even permutation of the triplet (1,2,3). Let us now consider the Coriolis group (1.1.5). In relation to it, the 2–forms Ω3 given by (1.1.1) are no longer invariant. Indeed, a direct calculation leads us to the 2–form: Ω 4 = mdv i ∧ dq i + ( E i dq i − mv i dv i ) ∧ dt + B i dq j ∧ dq k , (1.1.9)

called the expanded Lagrange form. In the expression of the 2-form Ω4, the coefficients Ei and Bi are functions of t, qi, vi and the summation is done after i = 1, 3 in the first three terms and after the even permutations (i, j, k) of (1,2,3) in the last term. The following two results are deduced through a direct calculation. Proposition 1.1.2 Forms (1.1.9) are invariant to group (1.1.5); fields E and B, as speed v, change according to the formulas

v = A( v − u ) B = A( B + 2 mR ) E = A( E − B × u − mΓ ) ,

(1.1.10) where R, u and G are respectively the instantaneous rotation vector, the entrainment speed and the entrainment acceleration. The expressions of the vectors u and G are given by the formulas:

dr Γ dt ⎡ ⎛ dR ⎞ d 2r 2⎤ Γ = ⎢ j⎜ ⎟ + ( j( R )) ⎥( q + r ) − 2 . dt ⎣ ⎝ dt ⎠ ⎦

u = j( R )( q + r ) −

(1.1.11) Proposition 1.1.3 If the dynamic system is characterised by equations such as (1.1.1), then the forms of family (1.1.9) lead to the same movement equations (1.1.1), if and only if the Lorentz condition is satisfied: F i = E i + ( v × B )i , i = 1,3. (1.1.12) Therefore, family (1.1.9) depends on only one arbitrary field, for example B; E being determined by (1.1.12) where Fi are given and vi are obtained from (1.1.1) through integration, under the given initial conditions. 3 Lagrange’s generalized form The fact that 2–form Ω4 imposes through Lorentz’s condition (1.1.12) a special structure upon field F leads us to replace it through another 2–form called generalised and having the expression: Ω 5 = mdv i ∧ dq i + ( E i dq i − P i dv i ) ∧ dt + B i dq j ∧ dq k + Q i dv j ∧ dv k , (1.1.13) in which coefficients E, B, P, Q depend on t, q, v. 2–form Ω5 is invariant to the Galilei group (1.1.2), its coefficients changing according to the formulas

B = A( B ),

E = A( E − B × u ),

P = A( P + u ), Q = A( Q ) ,

(1.1.14) where u represents the entrainment speed. Requesting to system (1.1.4) to be equivalent with system (1.1.1), in other words, form Ω5 induce the same Newtonian movement equations, implies that its coefficients must satisfy Lorentz’s conditions :

22



































F = E + v× B v = P − F ×Q .

(1.1.15) It implies that, out of the four coefficients E, P, B and Q of form Ω5, two (for example B and Q) are independent, the other two (E and P) are determined through equations (1.1.15), if the movement elements are known. Reciprocally, functions E, B, P and Q being given, formulas (1.1.15) allow the determination through explicitness of speed v and force F. In the hypotheses BQ ¹ 1, m = 1, keeping in mind that: (1.1.16) it is obtained that:

det( j( B ) j( Q ) + I ) = ( BQ − 1 ) 2 ≠ 0 ,

v = ( j( Q ) j( B ) + I ) −1 ( P + j( Q )E ) (1.1.17) or the equivalent:

F = ( j( B ) j( Q ) + I ) − 1 ( E − j( B ) P ) , v = P + j( Q )[( j( B ) j( Q ) + I ) −1 ( E − j( B )P )]

F = E − j( B )[ j( Q ) j( B ) + I ) −1 ( P + j( Q )E )] . (1.1.18) With these values, the evolution equations are written: (1.1.19) respectively:

dq = ( j( Q ) j( B ) + I ) − 1 ( P + j( Q ) E ) dt

d 2q = ( j( B ) j( Q ) + I ) − 1 ( E − j( B ) P ) , 2 m dt

(1.1.20) where j(Q)j(B) = [j(B)j(Q)]t and so j(Q)j(B) + I = [j(B)j(Q) + I]t .

4 Lagrange’ complete form A 2–form of type (1.1.13) is not invariant towards a rotation referential change (1.1.5). Direct calculation leads us to the 2–form Ω6, expressed as:

Ω = A dv i ∧ dq j + ( E dq i − P dv i ) ∧ dt + B i dq j ∧ dq k + Qi dv j ∧ dv k ,

6 ji i i (1.1.21) in the case of the three dimensional space, or more generally (in case of arbitrary dimensions):

1 1 Ω 6 = Aji dv i ∧ dq j + ( Ei dq i − Pi dv i ) ∧ dt + Bij dq i ∧ dq j + Qij dv i ∧ dv j , 2 2

(1.1.22) where Bij = – Bji and Qij = – Qji . The characteristic equation system associated to 2–form Ω6, given by (1.1.22), is:

Bij dq j − Aij dv j + Ei dt = 0 , (1.1.23)

A ji dq j + Qij dv j − Pi dt = 0 ,

to which their linear combination is added Ei dqi - Pi dvi = 0. System (1.1.23) is equivalent to system (1.1.1) if and only if:

⎛ Bij det ⎜⎜ ⎝ A ji

− Aij ⎞ ⎟≠0 Qij ⎟⎠

(1.1.24) and Lorentz’s conditions are fulfilled, becoming in this case:

Ei = Aij F j − Bij v j , (25)

Pi = A ji v j + Qij F j ,

as well as their linear combination Ei vi dynamics. The following relation takes place:

- Pi Fi

= 0, an identity that represents a fundamental law of

23

− Aij Qij Pj

Ei ⎞ ⎟ − Pi ⎟ = 0 0 ⎟⎠

.

1.1.2 MAXWELL’S PRINCIPLE As it is known, there are examples of very important dynamic systems, described by 2–forms of Ω4 type, closed: dΩ4 = 0. This condition was elevated to the rank of principle by Souriau and called Maxwell’s principle. Below we will study the manner in which the 2–form classes (Ω3,...,Ω6) taken into consideration verify Maxwell’s principle and we will infer certain properties that are specific to each one of them. 1 Maxwell’s simple principle Let us suppose that the dynamic of a material point of mass m, subjected to the action of a force F of components Fi in relation to an orthonormal Galilean coordinate system is described by a 2–form of type Ω3, given by (1.1.1) (whose rank is 6). Condition dΩ3 = 0 leads us to the equations

∂Fh ∂F j ∂Fi − h = 0 , ( rot q F = 0 ), =0. j ∂q ∂q ∂v h

(1.1.26) The last conditions tell us that F does not depend on the speed (Fi = Fi(t, q)), and the first, that for every local t, F is a gradient:

Fi = −

∂V ( t , q ) . ∂q i

Such a dynamic system is called of class (N3) and in this case F is a quasi-conservative force field. A subclass (N2) is constituted by given dynamic systems through conservative fields (F= F(q)). In turn, class (N2) admits subclasses (N0) Ì (N1) Ì (N2) where (N1) corresponds to the case F = constant (uniform accelerated systems) and (N0) contains the inertial movements (case F = 0). The condition dΩ3 = 0 is named Maxwell’s simplel principle. 2 Maxwell’s expanded principle A 2–form of type Ω3 is not always closed, an example being the movement of a projectile in the atmosphere, for which the air resistance force is a force complicated by speed. We are thus led to consider the family of 2–forms of type Ω4 (of rank 6) given by (1.1.9), where field F is given, B is arbitrary and E is given by (1.1.12). Obviously, rank W4=6. By imposing Maxwell’sexpanded principle (dΩ4 = 0) on 2–form Ω4, the well-known electro-dynamics Maxwell equations are obtained:

∂E i ∂B i ∂B = j = 0, j ∂v ∂v rotqE + ∂t = 0, divqB = 0.

(1.1.27) By keeping in mind the relations (1.1.12) it immediately follows that: Proposition 1.1.4 A necessary condition for the Ei and Bi functions to exist, with the verification of ⎛ ∂F i ⎞ ⎜ ⎟ ⎜ ∂v j ⎟ ⎝ ⎠ to be anti-symmetrical and for its elements not to relations (1.1.27) and (1.1.12), is for matrix depend on speeds. Demonstration Indeed, if relations (1.1.27) take place, then B and E do not depend on speed and they result based on (1.1.12) (1.1.28) Fi = Ei (t, q) + (vj Bk (t, q) – vk Bj (t, q)) , where (i,j,k) represent an even permutation of (1,2,3). Obviously, 2

i

∂ F ∂F i ∂F j = − =0. j i ∂v ∂v and ∂v j ∂v k We will note with (N4) the class of dynamic systems that can be described through pre-symplectic 2– forms of type Ω4. Obviously (N3) Ì (N4). 3 Maxwell’s generalized principle As it can be seen from the examples that are given at the end of this section, not all dynamic systems belong to the (N4) class. By imposing Maxwell’s generalized principle to a 2–form of Ω5 type given by (1.1.13), the following equations are obtained:

24

 



























⎛ Bij ⎜ det ⎜ A ji ⎜− E j ⎝













∂B i = 0, ∂v h ∂Q i = 0, ∂q h

∂E i ∂P h − =0. ∂v h ∂q i

(1.1.29)

We will use (N5) to denote the class of dynamic systems that can be described through presymplectic 2–forms of type Ω5 and we have (N4 ) Ì (N5 ). 4 Maxwells complete principle Let us now consider the most general case, in which the dynamic of a system is described through a 2–form of Ω6 type, whose local expression (in a variety map, product R × TM , dim M = m is given by (1.1.22), and whose coefficients satisfy Lorentz’s relations (1.1.25). Maxwell’s complete principle (dΩ6 =0), imposed to the 2–form Ω6, leads us to the equations:

a) c) e)

∂Qij h

∂q ∂Aij ∂t

∂Qij ∂t

+

∂Ahj ∂v

i

+

∂Ei ∂v j

+

∂Pi ∂v j

∂Bij ∂Aih ∂A jh ∂Ahi = 0, b ) + − = 0, j ∂v ∂v h ∂q j ∂q i ∂Pj ∂Bij ∂E j ∂Ei + i = 0, d ) + i − j = 0, ∂t ∂q ∂q ∂q ∂Pj ∂Bij ∂Qij − i = 0, f ) ∑ = 0 , g ) = 0, ∑ k k ∂v ( i , j ,k ) ∂q ( i , j ,k ) ∂v −

(1.1.30) where the sum is carried out by the circular permutations of permutation (i,j,k) in the last two relations (1£i = du b ⎜ a ⎟ = δ ab < Di ⎜ a ⎟ , du b > a ∂u ⎝ ∂u ⎠ ⎝ ∂u ⎠ we deduce

From the compatibility condition, ∗ ∂ ∂ ∗ ∂ < a , Di ( du b ) > < M ca c , du b > < a , M bc du c ) > δ cb M ca + M bc δ ac i i i + ∂u = i ∂u + ∂u = , relationship from which it follows that the matrices We have

M i

a b

∗

and

M i

a b

are opposing one another.

Di ( ω ) = ( Di ( Ya ) − Yb M i

b a

)du a

. We note with:

241











i

∗

= Di ( Ya ) + Yb M

ΔYa

b a

i

= Di ( Ya ) − Yb M

b a

(10.1.17) Δx , or Δx . These functions are called covariants partial derivatives of the Ya components of the ω form in the given map. On a local map change, we have the formulas: ΔYa ∂u α ΔYα = Δx i ∂u a Δx i . ΔYα i The Δx functions are therefore the components of a covector. The functions Di Ya are, for the time i

i

being, unspecified (outside of relationships (10.1.2'). Observation From the covariance condition of the derivatives expressed through (10.1.4), we have for ∗

M i

a b

formulas of transformation equivalent to (10.2.3’), considering the anti-transposition relationships. 4 Let’s suppose now, in general, a d-tensor B of order q, q times covariant, written locally as:

B = Bb1 ....bq du b1 ⊗ ... ⊗ du

bq

. We have: q

∗

k =1

i

Di B = ( Di Bb1 ....bq + ∑ Bb1 ...bk −1ck bk + 1 ....bq M where functions We note

Di Bb1 ....bq

ck bk

)du b1 ⊗ .... ⊗ du

bq

,

are unspecified, outside of relationships (10.1.3). ΔBb1 ....bq Δx

i

q

∗

k =1

i

= Di Bb1 ....bq + ∑ Bb1 ...bk −1ck bk + 1 ....bq M

ck bk

,

B functions we call partial covariant derivatives of the components b1 ....bq On a local map change, they change by formulas: β ΔBb1 ....bq ∂u β1 ∂u p ΔB β1 ....β q = .... b Δx i Δx i , ( ∀i = 1, n ). ∂u b1 ∂u p 5 Being given now a field of tensors T of p+q order, p times contravariant and q times covariant. We will locally write it as: ∂ ∂ a ...a b T = Tb1 1....bqp ⊗ .... ⊗ a ⊗ du b1 ⊗ .... ⊗ du q a1 p ∂u ∂u . By applying the operator Di , these functions shall be obtained: a ...a

ΔTb1 1....bqp i

p

a ...a

a ...c ....a

h p = Di Tb1 1....bqp + ∑ Tb1 1.......... ...bq M

ah ch +

Δx h =1 called partial covariant derivatives of the T-tensor components . On a local map change, they change by the formula: i

a ...a

1 p ΔTb1 .... bq

Δx i

=

∂u a1 ∂u α 1

....

∂u

ap

∂u β1

∂u

αp

∂u b1

....

q

a1 ...........a p

∑ Tb ....c ....b

k =1

1

k

q

∗

M i

ck bk

,

α ....α

∂u

βp

ΔTβ11....β q p

∂u

bp

Δx i

, ( ∀i = 1, n ).

a ...a

Functions

Di Tb1 1....bqp

are unspecified, outside relationships (10.1.2) and (10.1.3).

10.1.4 Calculation of operators Di B) Existence of derivation operators 1 Calculation of values of operators Di on components Xa Proposition 10.1.9 Relationships (10.1.2') define operators Di (in fields).

242



















ΔYa













c Indeed, let's consider, for c fixed, the property that u can be regarded as the parameter on a curve ∂u α u α = u α ( u c ) , and ∂u c as the field of vectors tangent to the curve, u α = u α ( u c ) and reciprocally, any field of vectors is a tangent field to a (congruence of) (integral) curves in another map, so that in general: dX b ε aj a j ib dx (10.1.18) D i X = ,

which justifies the assertion of proposition 10.1.1, so that: ⎡ αj dX β ΔX α ∂ α β ⎤ ∂ ε + M X = ⎢ iβ ⎥ α β i dx j Δx i ∂u α ⎦ ∂u (10.1.18') Di (X ) = ⎣ . a b ΔX dX = aj + M ba X b i ε i Δx dx j ib functions, called partial covariant derivatives of components Xa, in the given map, are specifically expressed with the values of operators Di on components Xa, given by (10.1.1). On a map change, they change by formulas (10.1.1'): dX b ∂u a αj dX b a b a b ε ib ε aj j +M bX α j +M bX i i dx ib dx = ∂u , which informs that partial covariant derivatives of components of a d-field of vectors constitute d-fields of a vectors. Formulas (10.1.1) specify D i X functions.

[

]

α α h 2 Being given, on manifold M, a submanifold k (through equations as u = u ( x ) ), we will say that the d-field X is transported through parallelism on submanifold k, reported to the given connection if along this submanifold all its partial covariant derivatives are annulled: The conditions of transport through parallelism are now: dX b a b ε ibaj j +M bX i dx (10.1.19) = 0.

X ia =

∂u a ∂x i vectors are transported by

A submanifold ua=ua(xh) is called self-parallel if its tangent parallelism, that is, if: ∂u b ∂ 2u b + M ba h aj h j i ∂x (10.1.20) = 0. ε ib ∂x ∂x ( i , j , h = 1, n ). 3 The extension of operator Di to d-1-forms Relationship (10.1.5) is fulfilled based on formula (10.1.4). One of its solutions is obtained through the formula: dY Di ( Ya ) = ε iabj bj dx . With these formulas, the partial covariant derivatives of components Ya of a d-(1-form) are: ΔYa dY = ε iabj bj − Yb M ba i i Δx dx . 4 Extension of Di operators to functions 1 τ i j = ε iaaj aj m a) Being given the mixed tensors ε ib , with their help we can construct the functions . j

Proposition 10.1.10 Functions τ i are invariant to changes of local maps on M. ∂u a ∂u β αj τ i j = ε iaai = α ε = δ αβ ε iαβj = ε iααj = mτ i j a iβ ∂ u ∂ u Indeed, m .

243





b) Being given an X field, of components Xa and an f function. Being given the field Z = fX, of fXa df dX b Di Z a = ε ibaj ( f + j Xb ) j dx dx component. By applying the operator Di , we have: as well as a a b Di Z = Di ( f )δ b X +

+ fε ibaj

dX b dx j . Identifying the two expressions we have:

df dx j . i) From formula i) retrieve, by contraction, df Di ( f ) = τ i j dx j , ii) any function that fulfills relationships i), fulfills relationships ii) too. We will retain only those distinguished functions f, which fulfill relationships i). Their set is a 1 subalgebra of the algebra of all functions defined on j n ( M ) . This subalgebra, is determined by the Di ( f )δ ba = ε ibaj

SD

geometric structure considered, and is denoted by: F d . a c) Being given now a d-(1-form) ω = Ya du . Let's consider d-(1-form) fω, of components fYa . Using operator Di, it shall be obtained: di(FYa) = di(f) Ya+ fDi(Ya), as well as dYb ⎤ ⎡ df Di ( fYa ) = ε iabj ⎢ j Yb + f ⎥ dx j ⎦ . Comparing the two formulas we obtain the same expression found above. ⎣ dx

M

a b

N

a b

C) Simplified writing Let’s replace coefficients with others: , given through connection a aj c M b = ε ic N b j formulas: i , substitution whose motivation will be seen below. N ba 5 Partial covariant derivatives expressed using coefficients i . 1a Partial covariant derivatives of the components of a vectors field. Being given a field of vectors ∂ X =Xa a ∂u . Partial covariant derivatives of components Xa have X written in a local map in the form of: been given by formula (10.1.1'): c c aj ⎛ dX c b⎞ aj δX ΔX a dX b ⎜ ⎟ a b ε + N X = ε ic ⎜ b ic = ε aj j ⎟ j j +M bX δx j ⎝ dx ⎠ i Δx i ib dx = , c c δX dX = + N bc X b j j j dx where the functions δx will be called short partial covariant derivatives of components Xa . In relation to a map change, they change by formulas: δX a ∂u a δX α = α δx i ∂u δx i . Proposition 10.1.11 A field of vectors X is transported by parallelism if: dX γ + N γβ X β = 0 j i (10.1.21) dx , A submanifold is self-parallel if: b ∂ 2u a 1 ⎛ a ∂u b a ∂u ⎞ ⎜N b ⎟ = 0. + + N b j ∂x i ∂x j 2 ⎜⎝ i ∂x j ∂x i ⎟⎠ ( ∀i , j = 1, n ) (10.1.22) . It results from (10.1.2) and (10.1.3) respectively. i

244

i





















a

2a Partial covariant derivatives of a d-1- form Being given a d-(1-form) ω = Ya du . Partial covariant derivatives of its components are given by the formula: (10.1.1'): ΔYa δY ⎡ dY ⎤ δYb dYb = ε iabj ⎢ bj − N cn Yc ⎥ = ε iabj bj = − Yc N cn i j j Δx δx , cu δx j dx j ⎣ dx ⎦ . δYb dYb = j − Yc N cn j j dx We call the expressions δx , short covariant partial derivatives. On a map change, they change by formulas: δYa ∂u α δYα = δx i ∂u a δx i . M an Observation If in the change formulas, on a map change, of coefficients i , we notice their N bc character apparently geometric, in the ones of j their “dynamic” character is evident. Di ( du a ) = −ε icaj N bc du b ρ i = ( Di ( Ya ) − ε icbj N ca Yb )du a j j Based on previous notations, we have: . Compatibility condition 2 requires fulfilling these relationships: ∂ ∂ ∂ Di ( du a ( b )) = Di ( du a ) b + du a Di ( b ) = M ba − ε icaj N bc = 0 i j ∂u ∂u ∂u . formula from which we have N ba Proposition 10.1.12 On a local map change, the functions i change, based on formulas (10. 2.3) a M n of coefficients i change, by the formulas: ∂u β ⎤ ∂u b ⎡ ∂u α N ba = β ⎢ a N αβ + ia ⎥ i ∂u ⎣⎢ ∂u i ∂u ⎦⎥ (10.1.23) , and reciprocally. 3a Partial covariant derivatives of a tensor field By extension, we can now consider short partial derivatives, for the components of an arbitrary tensor field T of the order p + q, p times contravariant and q times covariant, of the form: ∂ ∂ a ...a b T = Tb1 1....bqp ⊗ .... ⊗ a ⊗ du b1 ⊗ .... ⊗ du q a1 p ∂u ∂u . Short partial covariant derivatives can be given by the formula: a ...a

δTb1 1....bqp i

a ...a

=

dTb1 1....bqp i

p

a ...c ....a

h p + ∑ Tb1 1.......... ...bq N

δx dx h =1 On a local map change, they changed by the formula: a ...a

1 p δTb1 .... bq

δx i

=

∂u a1 ∂u α 1

....

∂u

ap

∂u β1

∂u

αp

∂u b1

....

i

ah ch −

q

a1 ...........a p

∑ Tb ....c ....b

k =1

1

k

q

N i

ck bk

,

α ....α

∂u

βp

δTβ11....β q p

∂u

bp

δx i

, ( ∀i = 1, n ).

1 10.1.5 Coreferences and references adapted on the jets maifold J n ( M ) 1a Adapted coreferences Let’s construct, with the natural coreference, the independent Pfaff forms: aj a a b b a a a a b δu i = M b du + ε ib du j δ u = ε du , i b (10.1.24) , ( ε b = δ b , prop .1.1.1 ) On a local map change, the covectors of this coreference change by formulas: ∂u a ∂u a δu a = α δu α , δu ia = α δu iα ∂u ∂u (10.1.25) . o 2 Adapted references Let’s suppose now the dual adapted reference as:

245





















δ ∂ ∂ δ ∂ = ε~ab b − N ba b , = ε~ jabi b . a a j δu ∂u ∂u j δu i ∂u j

(10.1.26) Proposition 10.1.13 The reference (10.1.3) is dual to the coreference (10.1.1) if: ε ba ε~cb = δ ca , ( ε~ba = δ ba ), ε ibaj ε~ jcbh = δ ca δ ih , M ba − ε icaj N bc = 0 , N ba = ε~icbj M ca . j j j (10.1.27) i ε~ ai Functions jc are defined by relationships (10.1.42) and duality conditions: ⎛ δ ⎞ ⎛ δ ⎞ ⎛ δ ⎞ δu a ⎜ b ⎟ = δ ba , δu a ⎜ b ⎟ = 0 , δu a ⎛⎜ δ ⎞⎟ = 0 , δu ia ⎜ b ⎟ = δ ba δ i j i ⎜ δu j ⎟ ⎜ δu j ⎟ ⎜ δu ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ δu ⎠ , imply relationships (10.1.4). N ba Proposition 10.1.14 On a local map change, functions i change, based of formula (10.2.3)of a M b change of coefficients i , through formula (see also proposition 10.1.4): c ⎞ a ⎡ d ⎛ ⎤ a ∂u ia ⎜ N ba = ∂u ⎢ ∂u N cd + ∂u i ⎥ ⎟ ∂u c a ∂u c c b b N = N + ⎜ i c b ∂u ⎢⎣ ∂u i ∂u ⎥⎦ ⎟⎠ b ∂u c i ∂u b , ⎝ (10.1.28) ∂u i . By a change of local map, the vectors of the adapted base are subject to the passage formulas:

δ ∂u b δ δ ∂u b δ = , = a ∂u a δu b δu ia ∂u a δu ib (10.1.29) δu

.

⎛ δ ⎞ ⎜ ⎟ ⎜ δu b ⎟ ⎠ y generate a It follows that vectors generate the vector space Vy, and vectors ⎝ T J 1M = H y ⊕ Vy complementary space Hy, called the horizontal space, so that y n . b M a From the formulas (10.1.6) it follows that, given a connection ( i ), the vertical V distribution, δ δu α Of local mn dimension, generated by fields i , decompose in n vertical distributions Vi , each ⎛ δ ⎜ ⎜ δu a ⎝ i

⎞ ⎟ ⎟ ⎠y

(

)

n

of dimension m (fact due to the

i a ab

metrics’ existence ), so that:

V = ⊕ Vi

as well as an H horizontal δ a distribution, of dimension m, generated by vectors: δu . ⎛ δ δ ⎞⎟ ⎜ ⎜ δu a , δu a ⎟ i ⎠ The module S generated by fields ⎝ can now be presented thanks to the existence of b M a connection ( i ), as direct sum: S = H ⊕ V , decomposition with which we identify the connection existence. N ba ~ a ,ε~ aj ε o b ib 3 Obtaining coefficients ( şi i ) i) To obtain these coefficients, based on the 10.1.1 sentence and the formulas (10.1.41), it follows: a ~ ε b = δ ba . Consequently, the systems (10.1.1) and (10.1.3) get the form: δu a = du a , δu ia = M ba du b + ε ibaj du bj , i (10.1.24')

246

1







(10.1.26 ')

δ ∂ ∂ = a − N ba b a i δu ∂u ∂u i

,

δ ∂ = ε bija b a δu i ∂u j

.

~a ii) The conditions (10.1.42) of definition of functions ε b constitute a system of linear and nonhomogeneous equations, a system which can be written in matriceal form: ⎛ ε 1ac1 ε 1ac2 .... ε 1anc ⎞⎛ ε~1cb1 ε~1cb2 . ε~1cnb ⎞ ⎛ δ ba 0 . 0 ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ε 2ac1 ε 2ac2 .... ε 2anc ⎟⎜ ε~2cb1 ε~2cb2 . ε~2cnb ⎟ ⎜ 0 δ ba . 0 ⎟ ⎜ ⎟⎜ ⎟=⎜ ⎟ . . . ⎟⎜ . . . . ⎟ ⎜ . . . . ⎟ ⎜ . ⎜ ε a1 ε a 2 .... ε an ⎟⎜ ε~ c1 ε~ c 2 . ε~ cn ⎟ ⎜ 0 0 . δ ba ⎟⎠ nc nc ⎠⎝ nb nb nb ⎠ ⎝ nc ⎝ . The following properties are highlighted: aj ah ~ cj a j ~ aj 1o Matrices ( ε ib ) and ( ε ib ) are both proper (invertible), hence the system ε ic ε hb = δ b δ i is a Krammer system and, therefore, it accepts a unique solution. aj ~ aj 2a Matrices ( ε ib ) and ( ε ib ) are inverse to one another. a j ~ ah cj 3o Transposing the matriceal form, we get the relationships: ε ic ε hb = δ b δ i . ε~ ai On a local map change jc coefficients change through the formulas: ∂u a ∂u β ~ αj ε~ibaj = α ε iβ ∂u ∂u b . a ~ ah c N b = ε ic M b h iii) The inverse of formulas (10.1.43) are: i Based on these properties, the systems (10.1. 1 ') and (10.1.3') may be inverted and we obtain respectively the systems: du a = δu a , du ia = ε~ibaj ⎛⎜ δu bj − M bc δu c ⎞⎟. j ⎝ ⎠, (10.1.24 ")

∂ δ ∂ δ ∂ δ δ c δ = a + ε bh , = ε bija b , = a + M ca b jc N a a b a a h i ∂u δu δu j ∂u i δu j ∂u δu δu i (10.1.3 ") ( ) ai ~ ε with jc functions defined above. δ ⎤ ⎡ δ ⎢ δu a , δu b ⎥ ⎦ Lie 10.1.6 Important geometric objects 1o Being given the adapted base (10.1.3) and ⎣ crochets associated, we have the property: δ ⎤ ∂ ⎡ δ c δ δ R cab = b ⎛⎜ N ca ⎞⎟ − a ⎛⎜ N bc ⎞⎟ ⎢ δu a , δu b ⎥ = Rj ab ∂u c ⎣ ⎦ j δu ⎝ j ⎠ δu ⎝ j ⎠ , , where: j formulas that, expressed in the adapted reference, become: δ ⎤ ~c δ ⎡ δ ~ d ⎢ δu a , δu b ⎥ = Rj ab δu c R cab = ε ch jd R ab ⎣ ⎦ j h j , where . c R ab The j object will be called tensor of curvature, and the object: 1 ∂ R = R cab du b ∧ du a ⊗ c 2 j ∂u j , form of curvature. ∂ N bc ⎡ ∂ δ ⎤ ∂ j ⎢ a , b⎥=− a ∂u δu ⎦⎥ ∂u i ∂u cj 2o Analogously, we have the crochet: ⎣⎢ i with which we form the object:

247

















⎛ ∂ N c ∂ N bc ⎞ ⎡ ∂ ⎟ ∂ δ ⎤ ⎡ ∂ δ ⎤ ⎜ j a j i i i ∂ − cj δ c d ⎢ a , b ⎥−⎢ b , a ⎥=⎜ b a ⎟ c t = t ε = τ ab ab hd ∂u i ⎟ ∂u j ⎣⎢ ∂u i δu ⎦⎥ ⎣⎢ ∂u i δu ⎦⎥ ⎜ ∂u i j ∂u cj j δ hc h ⎝ ⎠ = This object is called torsion, and the forms: ⎛ ∂ N c ∂ N bc ⎞ ⎟ b 1⎜ j a ∂ j δ − du ∧ du a ⊗ c 1 i c ⎜ b a ⎟ τ ab du b ∧ du a ⊗ c 2 ⎜ ∂u i ∂u i ⎟ ∂u j 2h δu h ⎝ ⎠ = , torsion forms.

c ab

δ δu hc

.

10.1.7 Projection operators: vertical v and horizontal h, associated to a connection 1 Being given a field of X vectors on J n ( M ) which, in a natural map, is written in the form (10.1. 7)

X = Xα

∂ ∂ + X iα α α ∂u ∂u i

. Through a reference change of form (5.4), X becomes: δ δ c a X = X a a + ε bh + X hc jc N a X h δu δu bj . It has a vertical component vX in V and a horizontal hX component in H, so that X = vX + hX. So we have: δ c a vX = ε bh + X hc , hX = X a δ , jc N a X b h δu j δu a

(

)

)

(

that makes the transition from the adapted to the natural reference. Rewriting these components in the natural base, we have: ⎛ ⎞ ∂ a⎜ b ∂ b ∂ ⎟ hX = X ε − N . vX = N ca X a + X hc , a a ⎜ ∂u b j h ∂u bj ⎟⎠ ∂u bj ⎝ If X is vertical then hX = 0, if X is horizontal, then vX =0. Proposition 10.1.15 We have the properties: vvX=vX, hhX=hX, vhX=0, hvX=0.

(

)

§ 10.2 SYSTEMS OF EQUATIONS WITH PARTIAL DERIVATIVES AND ASSOCIATED GEOMETRIC STRUCTURES 1 Systems of equations with partial derivatives of first order, implicit. Given M = Mm a differentiable manifold of a finite size m, J1(Rn, m) the manifold of first-order jets from Rn to M and fibrate space J1(Rn,M)´mT*M. ∗ a A function F:(x, u, v)ÎJ1(Rn,M) → Tu M which, in local writing with x = (xi), u = (ua) and v = (u i ), shall be illustrated by:

a a i b b ∗ F: (xi, ua, u i ) → F(xi, ua, i ) = fa( x , u , u i )duaÎ Tu M . Definition 10.2.1 The nucleus of the F function is given, locally, by the condition: i b b (10.2.1) F ( x , u , u i ) = 0,

a

where the functions of Fa are the components of a d-covector called system of equations with partial derivatives of the first order, implied. ⎛ ∂F ⎞ det ⎜⎜ ab ⎟⎟ ≠ 0 , ( ∀i ∈ 1, n ) ∂u We will assume fulfilled the condition of nondegeneration ⎝ i ⎠ . a a b From the definition of functions Fa, it follows that, at a local map change on M, as u = u ( u ) , functions Fa change by formulas: ∂u b Fa = a Fb ∂u (10.2.2) .

248



































Definition 10.2.2 We call solution of equations (10.2.1), a system of functions as: ua = ua(xi), with the property that, entered in equations of the system, with their partial derivatioves transform the equations (10.2.1) in identities: ∂u b ( x h ) x i , u b ( x i ), ) ≡ 0 , ∀( x i ) ∈ R n i ∂ x Fa( . Equations (10.2.1) define J1(Rn,M) a submanifold K, manifold of system solutions. From the definition of the solutions, it follows that a solution determines a submanifold k, whose lift K is located, until identification, in the core of function F, expressed by equations (10.2.1). 2 Systems of equations with elementary partial derivatives (scleronome). Forms with variations Definition 10.2.3 We will say that a system of equations with partial derivatives is elementary (scleronome) if it has the form: b b (10.2.3) F ( u , u i ) = 0, a

Let's consider the local map change on J1(R

n,

M), of the form (10.1.1). A vector field a

dξ ∂ ∂ ∂ ~ , X =ξ (u ) a X =ξ a a + + dx i ∂u ia generates on ∂u , on M with the extension ∂u pseudogroup with a parameter, given by formulas: a

b

u a = u a + εξ a , u ia = u ia + ε

J1(Rn,

M) the

dξ a , dx i

and with them, the given system is associated with variations forms: b i dξ a abξ b + a ab X dx i , (10.2.4) Ma = where: i a ab =

∂Fa ∂u ib

, a ab =

∂Fa ∂u b

,

i (10.2.5) . det( a ab ) ≠ 0 , ∀i = 1.n , which constitutes a system of forms with variations, associated with the given system (10.2.1). i A first geometric object, associated with the system (10.2.1), is the function assembly ( a ab ). On a local map change on M, these functions are changed by formulas (10.1.8): ∂u α ∂u β i a i a b αβ (10.2.6) a ab = ∂u ∂u , i and therefore, the functions a ab constitute a set of n "metrics". i i From the non-degeneration property, of coefficients a ab we get that, whichever i, matrices ( a ab ) can be inverted, the components of the inverse changing at a local map change, by formulas (10.2.1); they allow the construction of the coefficients defined by (10.2.5). A second geometric object is that of functions ( a ab ) which, on a local map change, change by

formulas:

⎡ ∂u β ∂u iβ i ⎤ ∂ 2uγ a + Fγ ⎢ b aαβ + ⎥ αβ ∂u ∂u b ∂u a ∂u b ⎢ ⎥ ⎣ ⎦ (10.2.7) . On the manifold K, of solutions of (10.2.1) (Fa= 0), change formulas get the form (10.1.11), of Christoffel’s coefficients of the first kind. a ab =

∂u α ∂u a

a ab =

∂u α ∂u a

⎡ ∂u β ∂u iβ i ⎤ a + a αβ ⎥ ⎢ b αβ ∂u b ⎣ ∂u ⎦.

(10.2.7 ") We can now construct functions of the form (10.1.12). M a Proposition 10.2.4 Coefficients i b define a geometric object on J1(Rn,M) and a connection on K. Indeed, at a local map change on M, they change to J1(Rn, M) through formulas:

249

















⎡ ∂u β ∂u β α ⎤ ∂u a ⎢ j ε 1 ∂u a ∂u c ∂ 2 u ε ~ αγ M ⎥ aj b a F b M a i ε α ⎢ ∂u ∂u i β ⎦ ib n ∂u α ∂u γ ∂u b ∂u c (10.2.8) i b = ∂u ⎣ + +, and respectively the manifold K by (10.1.13). ⎡ ∂u β ∂u β α ⎤ ∂u a ⎢ j ε M b b i β ⎥ aj M a α ⎢ ∂u ⎦, ib + ∂u (10.2.8 ') i b = ∂u ⎣ The connection defined by the formulas (10.2.8 ') is associated, canonically to the equation system (II. 1.1) and, as such, implies, with the same calculation, the existence of a partial covariant derivative Di in ∂ Λ = λi i λi D i . We ∂x direction from Rn :D = direction i and, by linearity, of a derivative along a certain have: Theorem 10.2.5 The lift of a self-parallel submanifold that has a point situated on the submanifold of the solutions, is entirely situated on this submanifold (is solution of equations (10.2.1)). Demonstration Being given a self-parallel submanifold, fulfilling equations (3.4), in which ε aj M a i coefficients ib are i b are constructed with objects a ab and a ab given by (2.3). This system is equivalent ∂Fa ∂F du bj + ab du b = dFa = 0 j b b b a u + a ab u h = 0 ∂u j ∂u with: ab jh , system that implies through contraction with dxh : . Theorems 10.2.7 (reciprocal to 10.2.5 theorem) Submanifold solutions of the system (10.2.1) are selfparallel submanifolds, in relation to N connection, associated to the system. Demonstration Being given ua = ua(xi), a solution of the system Fa = 0, i.e. fulfilling relationships ∂u b Fa ( u b ( x h ), h ) = 0 ∂x . By partial derivation, it follows:

(10.2.9)

c c ∂Fb ∂u j ∂Fb ∂u ∂ 2u c ∂u c j + = a + a =0 bc bc ∂u cj ∂x h ∂u c ∂x h ∂x j ∂x h ∂x h

a ibc

ε icaj

2

.

c

c ∂ u a ∂u + M =0 c i ∂x j ∂x h ∂x h , which means that the

By contraction with , it is obtained: submanifold Ua = u(xi) is self-parallel. Reciprocally A condition sufficient for a submanifold to be self-parallel, is to have the conditions (2.2.7), hence that submanifold is a solution of the system (2.10.2). A necessary and sufficient condition for the self-parallel submanifold to be the system's solution, is that matrix A = (aij) is nondegenerate, where aij a ab are, in turn, the matrices ( ij ) by hypothesis, nondegenerate. The definition 10.2.8 Two systems are equivalent if they accept the same solutions. The properties common to all equivalent systems are called dynamic. N a i Theorem 10.2.9 Functions a ab şi a ab have geometric character and functions i b have dynamic character. i Indeed, the a ab and a ab functions are associated with the equation system (4.1) and change with the b b system change. Given the equivalent systems Fa = 0 and G a = Aa Fb = 0 , det( Aa ) ≠ 0 . Given the ∂G ∂A c ∂F ∂G ∂A c ∂F i bab = ba = ab Fc + Aac cb bab = ba = ab Fc + Aac cb ∂u ∂u ∂u ∂u i ∂u i ∂u i functions and . On the manifold of solutions, ∂ F ∂ F i i bab = Aac cb = Aac a cb bab = Aac cb = Aac a cb Nˆ a ∂u ∂u i they have the form: , . Let’s calculate the coefficients i b , we have: ˆ c A e a = a ad a = N a Nˆ a = b ac b a iad A d c eb i db i b i b . i cb = It follows that the connection coefficients are dynamic, they do not depend on the representative of the class of equivalence. c.c.t.d.

250



























2.3 Adjoint systems The system of forms with variations (10.2.4), associated to the system (10.2.1), is b b i dη ˆ ˆ a η + a Y ab ab ˆ dx i another system of forms with not unique; it depends on the X field . Being given M a = variations. X ˆ Y Definition 10.2.10 A system M a is called adjoint of the system M a if there is a system of functions J i, so that for two fields of vectors X and Y to have:

i ˆ ( X ) = dJ M X (Y ) − M dx i . Proposition 10.2.11 Any system of forms with variations accepts a (unique) adjoint system, and the adjunction relationships are: i da ba ˆ a − a = − . i ab ba ˆi dx i (10.2.10) a ab + a ba = 0 , i ˆi The adjunction relationship is an involution. The matrices ( a ab ) and ( a ab ) are, respectively, antitransposed to one another, nondegenerate and nonsymmetrical. We will consider the inverse of the latter ˆ ab too: ( a i ).

The formulas (10.2.10), (10.2.6) and (9, 2.7) result in the formulas for changing geometrical objects on a map change: ∂u α ∂u β i aˆ i a b αβ ˆ ab a ∂ u ∂ u (10.2.11) = , α ⎡ β ∂u iβ i ⎤ ∂u ∂u ∂ 2uγ ˆ aˆ ab = a + aˆ ⎢ αβ F a b b αβ ⎥ ∂u ⎢⎣ ∂u ∂u ⎥⎦ + ∂u a ∂u b γ . (10.2.12) and on K manifold, respectively: ∂u iβ i ⎤ ∂u α ⎡ ∂u β ˆa ab = ˆ aαβ + aˆαβ ⎥ ⎢ ∂u a ⎢⎣ ∂u b ∂u b ⎥⎦ . (10.2.13) Thus, on the manifold K, these functions constitute Christoffel’s functions of the first kind of a new ˆ connection N . Let's now consider the functions: aj ˆ ac ˆ j n εˆ ib = a i , a cb with the fulfilling of properties similar to those in proposition 10.1.2. aj ˆ aj Proposition 10.2.12 Between the components ε ib and ε ib we have the connection relationships: aj ˆ aj ( ε ib ) = ( ε ib ) T, ( ∀i , j = 1, n ). a bcj = ( a iac a cbj )T = n( ε ibaj )T.

ca ˆ aj ˆ ac ˆ j Indeed, n ε ib = a i a cb = a i ˆ Let’s define the functions ( N connection coefficients ): ˆ a nM b ˆ ac ˆ (10.2.14) i = a i a cb . On a map change, we have the component change formulas:

∂u β ˆ ∂u a ∂u a ∂u c ∂ 2 u ε 1 αγ M ˆ a aˆ Fε M α α γ b c i ∂u b i i ∂ u ∂ u ∂ u ∂ u ∂ u n b (10.2.15) = + +, formulas that define a second connection on K by: ⎡ ∂u β ∂u β ˆ α ⎤ ∂u a ⎢ j εˆ M β⎥ ˆ b α j M a α ⎢ ∂u ∂u b i ⎦ (10.2.16) i b ∂u ⎣ = iβ +

251

α β

⎤ ⎥ ⎦

⎡ ∂u βj ⎢ b εˆ ⎢⎣ ∂u

αj iβ











Proposition 10.2.13 Between coefficients (

ˆ M i

a b

)=(

M

a b

i

M i

a b

and

)T +

ˆ M i

a ica

a b

we have the connection relationships: da bcj

dx j .

Nˆ connection is not part of the N + T connection class (is not constructed with the same group of

metrics as N).

i ˆi ˆ Let's now consider the assembly of the two geometric structures ( a ab , a ab ) and ( a ab , a ab ) structures, ˆ constructed, respectively, from a group of metrics and a connection. N and N connections are defined by ˆ a M ba M N ba Nˆ ba coefficients i and i b , (see formulas (10.2.8 ') and (10.2.16)), and coefficients i and i are defined by the above formulas and: ˆ a = εˆ aj Nˆ c M ba = ε icaj N bc M b ic b i j j (10.2.17) and i . a ˆ a M b M aj b ˆ aj ε i ib Taking into account definitions of and respectively i and ε ib formulas (10.2.17) are retranscribed in the form of: a ab = a acj N bc aˆ ab = aˆ acj Nˆ bc j j , respectively . ∂ ∂ X =Xa a Y =Y a ∂u fields and ∂u a . Suppose they are Being given a submanifold k and two vector transported, on k manifold, by parallelism, the first in relation to the connection N, the second in relation to ˆ connection N , hence: dX a dY a ˆ a b a b + N X = 0 + N bY = 0 b i i i dx i , dx . i i Given now the metrics a ab and the "invariant" a ab XbYa (with the significance of scalar product – or

angle – between fields X and Y).

d i a ab X bY a i Let’s evaluate the function: dx , we have: i i ⎛ ⎞ ⎛ i ⎞ ⎛ da ab ⎞ i c i ˆ c ⎟ b a ⎜ da ab i c i ˆ c ⎟ b a ⎜ da ab i i ⎟ b a ⎜ ˆ ˆ ba − a N − a N X Y − a N + a N X Y − a + a ac b cb a ac b bc a ab i i i ⎜ dx ⎟ ⎜ dx ⎟ ⎜ dx ⎟X Y i i i i ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ = = . i da ab i i − a ab + aˆ ba =0 i dx Based on relations (10.2.10), we have , it follows: Theorem 10.2.14 If on a submanifold k, two vector fields X and Y, are transported by parallelism, the ˆ first in relation to connection N, the second in relation to connection N , then the "angle" formed between

(

)

i ˆi them, in the order X, Y, is constant (relative to any of the metrics ( a ab ) and ( a ab ). Average connection, general formulas.

~ ~ N

a

Objects defined by (II. 4.5) and (II. 5.5) lead, through mediation, to the dynamic object i b , with the law of changing components: β ~ a ⎡ ∂u j ~ ∂u β ~ ~α ⎤ ~ ∂ u ~ ⎢ b ε αj N ⎥ N a α ⎢ ∂u ∂u b i β ⎦ iβ ⎣ i b = ∂u + , ~ 1 1 ~ ~ ~ aj N a aj N a N a ~ ~ aj i i 2 b b where = ( + i b ) and ε ib = 2 ( ε ib + ε ib ). This object is a connection, called an average j

connection, if the matrices

ε ia b

are symmetrical.

10.2.4 Self-adjoint systems

252





































Definition 10.2.15 A system of forms with variations, associated with a given system of equations with partial derivatives, is called self-adjoint if it coincides with its adjoint, that is if: i i a ab + a ba = 0,

a ab − a ba = −

i da ba

i da ab

. dx i dx i (10.2.18) A system of first-order self-adjoint is necessarily written in the main form. Indeed, the equations of ∂Fa c u ∂u cj the last group of (10.2.18) are linear in ij and, as such, its coefficients: must be null. The equations are written in the main form: i c b c (10.2.19) Fa = Aab ( u )u i + Aa ( u ) =0, =

i i ∞ with Aab , Aa Î C (J0(R n, M)), det( Aab )¹0, ∀i = 1, n. The conditions of the self-adjunction (10.2.18), translated for systems written in the main form, become: i ∂Aab ∂A ∂A = 0 , b − a = 0. ∑ i i c a ∂ u ∂u ∂u b (1)(20) A ab + a ba = 0, ( a ,b ,c ) It follows the restrictive condition m = 2p.

10.2.5 Lagrange's function. Euler – Lagrange-Birkhoff’s equations 10 Lagrange's function. Euler-Lagrange equations Being given a function L: J1(R n, M), written a

locally in the form: L=L(ua,u i ). As it is known, such function leads to the system of the Euler-Lagrange equations: ∂2L ∂2L ∂L b u + u b − a = 0. ij a b a b i ∂u ∂u ∂u i ∂u ∂u (10.2.21) i j In general, such a system is a second order system, it is called completely degenerate if: ∂2L ≡ 0 , ∀i , j = 1, n ∂u ia ∂u bj (10.2.22) , A, b = 1, m . It follows: Proposition 10.2.16 System (10.2.21) is a system of first order if and only if it is completely degenerate. From the conditions (10.2.21), by integration, it follows: ∂L = Aia , L = Aai u ia + B a ∂u i , i i a a where Aa = Aa ( u ), B = B( u ) . Proposition 10.2.17 The necessary and sufficient condition for a system of equations with partial a

derivatives of the first order to be Lagrangean is that Lagrange function to be linear in variables u i . 20 Euler – Lagrange – Birkhoff equations The equations (10.2.19) now become: ⎛ ∂Aai ∂Abi ⎞ b ∂B ⎜ ⎟ ⎜ ∂u b − ∂u a ⎟u i − ∂u a = 0 ⎝ ⎠ (10.2.23) . These equations are called Euler – Lagrange – Birkhoff’s equations. We will assume the hypothesis that whatever the index I is, we have: ⎛ ∂A i ∂A i ⎞ det ⎜ ab − ba ⎟ ≠ 0 ⎜ ∂u ∂u ⎟⎠ ⎝ , and the system (10.2.23) will be called ordinary. The equations Euler – Lagrange – Birkhoff are of the main form (2.4.2), with: (2.5.4)

253





















∂Aai

i Bab

b

−

∂Abi a

Ea = −

∂B ∂u a ,

= ∂u ∂u ) , (10.2.24 and fulfill conditions of the self-adjunction. On a local map change, coefficients of equations (II. 7.4) are changed by formulas: ∂u α ∂u β i ∂u α Babi = a B E = Eα αβ a ∂u ∂u b ∂u a (10.2.25) , . The connection coefficients are: ⎡ ∂B j ∂E ⎤ 1 a = Biac ⎢ cdb u dj + bc ⎥ b M n ∂u ⎥⎦ ⎢⎣ ∂u i .

Being given the set of systems of equations with first order partial derivatives, of the form (10.2.1), which meet the conditions required by the definition of 10.2.15. In this set it is considered the following equivalence relationship: Definition 10.2.18 Being given Fa = 0, and Ga = 0, two such systems. They are called equivalent if a a there is a matrix Ab (d-tensor distinguished) with the property that det( Ab ) ≠ 0, regardless point y on a 1

geometric area in J n M , so that:

( )

G a = Aab Fb , det Aab ≠ 0 . It follows from the definition of the equivalence relationship that all systems of such a class accept the same solutions. We will call a class of systems equivalent to the dynamic system, this notion equals the notion of system solutions. A notion or property is geometric, if it refers to the elements provided by a system of equations and, respectively, is dynamic if it is common to all equations in a class of equivalence. We have: i ab Theorem 10.2.19 The notion of metric, expressed by the functions: a ab , (resp. a i ) and Christoffel’s symbols of the first kind, expressed by the coefficients of forms with variations a ab , as those given by the ˆi ˆ ab aˆ ab system of adjoint forms a ab , a i ,, are geometric notions. They change when the representation of a M ba N ba i dynamic system is changed. Concepts: of connection (defined and expressed by its or i coefficients, ˆ a M Nˆ ba b respectively the one that is defined by coefficients i or i ), partial derivation operators Di, those of covariant derivation, of transport by parallelism and self-parallel manifolds, in relation to any of the two aj connections, symbols ε ib , objects of curvature and torsion, horizontal subspace, horizontal and vertical ~ components of a vectors X field are dynamic notions. Indeed, we have: ∂G ∂A c ∂F i bab = ba = ab Fc + Aac cb , ∂u i ∂u i ∂u i c ∂G ∂A ∂F bab = ba = ab Fc + Aac cb . ∂u ∂u ∂u i c i ab ac ˆ b On the manifold of solutions (Fa= 0) we have geometric properties bab = Aa a cb , bi = a i Ac , a ˆ a ac bab = Aac a cb , as the dynamic property εˆ ibaj = ε ibaj , Mi b = bi bcb = a iac a cb = Mi b , which confirms the dynamic character of the connection.

254

In this chapter we aim to associate to dynamic systems of the second order: superfields, whatever the generality of the equations through which these systems are defined and to describe the changing formulas of the field form components. § 11.1 LAGRANGE FORM This paragraph provides an overview of the superfield structure and presents the exchange formulas of the superfield components 11.1.1 Lagrange form. Geometric properties of a dynamic system A first geometric object, associated with a dynamic system, is Lagrange form. As stated above, to a dynamic Newtonian system, given by the kinematic equations: i j i j 11.1.0 q = F ( t , q , q )

, i

i

 we can associate a full "2-Lagrange form'' ( q = v ) [17]:

Ω = A ji dv i ∧ dq j + ( E i dq i − Pi dv i ) ∧ dt +

1 1 Bij dq i ∧ dq j + Qij dv i ∧ dv j , 2 2

11.1.1 The W form can be decomposed as follows: i∂ (11.1.1')W = w - ∂t WÙdt = w + qÙ dt. decomposition from which we have a first part:

A ji dv i ∧ dq j +

1 1 Bij dq i ∧ dq j + Qij dv i ∧ dv j , 2 2

(11.1.1 ' ') w= called reduced Lagrange form, as well as the component:

θ = E dq i − P dv i ,

i i (11.1.1 ' ' ') called generalized energy form (more specifically, generalized energy variation) associated to (11.1.1) and implicitly to the dynamic system (11.1.0), to which the form W is associated. We have: Proposition 11.1.1 The characteristic curves of 2-form (11.1.1), given by equations:

2-form

Bij dq j − Aij dv j + Ei dt = 0 , A dq j + Q dv j − P dt = 0 ,

ij i (11.1.2) ji are the very trajectories of the system (11.1.1), if and only if Lorentz's conditions are met:

Ei = − Bij v j + Aij F j , (11.1.3) and in addition:

Pi = A ji v j + Qij F j ,

⎛ Bij det ⎜⎜ ⎝ A ji (11.1.4)

− Aij ⎞ ⎟≠0 Qij ⎟⎠

. The equation: (11.1.2 ') Eidqi -Pi = 0, is a consequence of the equations (11.1.2). Lorentz's conditions delimit, in part, the arbitrary aspect of W form coefficients . Considering the system (11.1.1) of general form, with the field of non-null forces, to which a 2-form (11.1.1) is associated, fulfilling the condition (11.1.4), from the system (11.1.3) we express the functions of Fi as follows: a) If det (Aij)¹0, then from the first set of equations (11.1.3) we obtain:

255















































CHAP. XI GALLISSOT-SOURIAU’S FORMALISM ASSOCIATED TO ORDINARY DYNAMIC SYSTEMS OF SECOND ORDER

F i = A ih(EH+ BHJvJ ). b) If det (Qij)¹0, then from the second set of (11.1.9) we have: F i = Q ih(PH -AJHvJ). c) If both conditions from a) and b) are met, then the relationship obviously is: (AihBHJ + QihAJH) vJ = AihEH– QihPH. d) If nothing is known about the above conditions, then from the whole system (11.1.9) these functions are deducted: ΔF i Fi = , Δ where, with (11.1.4):

ΔF j =

Bij − A ji v j

− Ai 1 ,...,− Aij −1 ,− E j ,− Aij + 1 ,...,− Aim Qi 1 ,...,Qij −1 , Pj ,Qij + 1 ,...,Qim

(11.1.4 ') Equations (11.1.3) may also be written as matrix in the form:

⎛ Bij ⎛ Ei ⎞ ⎜⎜ ⎟⎟ = −⎜⎜ ⎝ − Pi ⎠ ⎝ A ji

− Aij ⎞⎛ v j ⎞ ⎟⎜ ⎟ , Qij ⎟⎠⎜⎝ F j ⎟⎠

⎛ v i ⎞ ⎛ Q ij ⎜ i⎟=⎜ ⎜ F ⎟ ⎜ − A ij ⎝ ⎠ ⎝

A ji ⎞⎛ E j ⎞ ⎟, ⎟⎜ B ij ⎟⎠⎜⎝ − Pj ⎟⎠

hence:

Aij Qij

.

Fi = -AjiEj - BijPj,

if:

⎛ Q ij ⎜ ⎜ − A ji ⎝

A ij ⎞ ⎟ B ij ⎟⎠

⎛ Bij ⎜ ⎜A = ⎝ ji

− Aij ⎞ ⎟ Qij ⎟⎠

−1

, ∂ ∂ ∂ + vi i + F i i ∂q ∂v . With the help of functions vi and Fi we obtain the spray S = ∂t Let's call, through abuse of language, solution of the system (11.1.1) a 2-form W , (11.1.1), fulfilling the relationships (11.1.3) and represent it through its own matrix (11.1.4). Observation While the 2-form (11.1.1) is defined through five groups of coefficients: (Aij, Bij, Qij, Ei, Pi), one of its solutions is defined only through three such groups (Aij, Bij, Qij) and describes the dynamics of (11.1.1). 11.1.3 Geometria of Lagrange’s 2-form W and the dynamic system (11.1.1) Behavior of Ω form coefficients on a map change i i h Let's consider a local map change on the manifold of M configuration: q = q ( q ) and respectively the vector map change on the tangent fibrate: ∂q i h i v = v i i h h q = q ( q ) ∂ q (11.1.5) , ; these functions differentials (of coordinates and speeds) change through the formulas: ∂q i ∂q i ∂v i dq i = h dq h dv i = h dv h + h dq h ∂q ∂q ∂q (11.1.5 ') , .

In view of these change formulas, the form W is given a similar form to (11.1.1), written in the new coordinates, from which, by identifying the coefficients, their transformation relationships are obtained: i

Ph =

Akh = 11.1.6

i

∂q ∂v ∂q i ∂q i ∂q j E = E − P , P , Q = Qij , h i i i hk ∂q h ∂q h ∂q h ∂q h ∂q k

⎞ ∂q i ⎛ ∂q j ∂v j ⎜ A + Qij ⎟⎟ , ji h ⎜ k k ∂q ⎝ ∂q ∂q ⎠

256



























,

Bij − A ji

which, inversed, lead to:





Δ=

⎛ ∂v i ∂q j ∂v i ∂q j ⎞ ∂q i ∂q j ∂v i ∂v j ⎜ ⎟ B + − A + ij ⎜ ∂q h ∂q k ∂q k ∂q h ⎟ ji ∂q h ∂q k Qij , ∂q h ∂q k ⎝ ⎠

In deducting these formulas, it was taken into account the antisymmetry of Bij coefficients ∂q i ∂v i = j ∂q j , deducted from (11.1.5). and Qij, as well as formulas ∂v The same result can also be obtained matriceally, through the components change of form (11.1.1) on a map change: ⎛ ∂q i ⎜ h ⎜ ∂q ⎜ ⎛ Bhk - Ahk Eh ⎞ ⎜ 0 ⎜ ⎟ ⎜ Qhk − Ph ⎟ ⎜ Akh ⎜ 0 ⎜ -E ⎟ ⎜ Pk 0 ⎠ k ( 1 1 . 1 . 6 ' )⎝ =⎝

⎛ ∂q j ⎜ k ⎜ ∂q ⎜ ∂v j ⎜ k ⎜ ∂q ⎜ 0 ⎜ ⎝

0 ∂v j ∂v k 0

relationship that represents the

∂v i ∂q h ∂v i ∂v h 0

⎞ 0⎟ ⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟ ⎠

⎛ Bij ⎜ ⎜ Aij ⎜− E j ⎝

− Aij Qij Pj

Ei ⎞ ⎟ − Pi ⎟ 0 ⎟⎠

⎞ 0⎟ ⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟ ⎠

In particular:

(11.1.6 ' ')

⎛ Bhk ⎜⎜ ⎝ Akh

⎛ ∂q i ⎜ h ⎜ ∂q − Ahk ⎞ ⎜ ⎟ 0 Qhk ⎟⎠ ⎜⎝ =

⎛ ∂q j ∂v i ⎞ ⎜ k ⎟ ⎜ ∂q ∂q h ⎟ i ⎟ ⎛ B Aij ⎞ ⎜ ∂v j ∂v ⎜ ij ⎟⎜ k ⎟⎜ ⎟ ∂v h ⎠ ⎝ − A ji Qij ⎠ ⎝ ∂q

⎛ ∂q i ⎜ h ∂q ( Eh ,− Ph ) = ( Ei ,− Pi )⎜ i ⎜ ∂v ⎜ h ⎝ ∂q

⎞ 0 ⎟ ⎟ ∂v j ⎟ ⎟ ∂v k ⎠ ,

⎞ o ⎟ ⎟ ∂v i ⎟ ⎟ ∂v h ⎠ .

(11.1.6 ' ' ') By identifying the coefficients, the relationships (11.1.9) of their change are obtained. Note The first two groups of equations in (11.1.9), are specific to form q , the last three, to form w. 11.1.3 Objects and geometric structures associated with form W a) From the fact that Pi = Pi(t, q, v), we deduce that to any vector v, of components vi, from point q, of coordinates qi, is associated the form of components Pi of the same point. The application (vi) ® (Pi) is called generalized Legendre transformation. From the transformation formulas (11.1.10), it is inferred that the Pfaff form: l = Pidqi, called Louville's d-form, or impulse form or kinetic form, constitutes a first important geometric object. Its external differential: ∂P ∂P 1 ⎛ ∂Pj ∂Pi ⎞ i ⎜ − j ⎟⎟dq ∧ dq j − ij dq i ∧ dv j − i dv i ∧ dt i ⎜ 2 ∂q ∂t ∂q ⎠ ∂v dl = ⎝ is generally non-null.

⎛ ∂Pj ∂P det ⎜⎜ i − ij ∂q ⎝ ∂q Proposition 11.1.2 If Pi = Pi(qH) and

symplectic structure on M.

257

























Bhk =

⎞ ⎟⎟ ≠ 0 , ⎠ and if M is orientable, then dl defines a























By abuse of language, its image on the cotangent manifold: dPiÙdqi is a d-form symplectic of the manifold T * M. Proposition 11.1.3 The external differential d l is null if and only if l is a closed basic form (its ∂V − i components do not depend on speed and time and derive from a potential function V = V(q), Pi = ∂q .) b) The Pfaff q form , defined by: q = Eidqi - Pidvi, called the derivative of the generalised energy system (energy derivative). c) Form: 1 s = 2 QijdqiÙdqj, characterizes the system's non-Lagrangean property. We have the relationship: ∂Qij i 1 1 ∂Qij i 1 ∂Qij i dσ = dq ∧ dq j ∧ dq h + dq ∧ dq j ∧ dv h + dq ∧ dq j ∧ dt . ∑ h h 6 ( i , j ,h ) ∂q 2 ∂v 2 ∂t . Proposition 11.1.4 The differential d s is null if and only if Qij are functions only of qh and ∂Qij =0 ∑ h ( i , j ,h ) ∂q . d) The geometric object (Aij, Qij) creating 2-form: 1 t = 2 (Aij - Aji)dq iÙdq j + Qijdv iÙdq j, is a tensor on M, distinguished in the second index and ordinary in the first (semidistinguished). The external differential of form t is: ∂Aij i 1 ∂A[ ij ] dτ = dq ∧ dq j ∧ dq h + ∑ dq i ∧ dq j ∧ dt 6 ( i , j ,h ) ∂q h ∂t + +

1 ⎡⎛ ∂Aij ∂A ji − ⎢⎜ 2 ⎣⎢⎜⎝ ∂v h ∂v h

⎞ ⎛ ∂Q jh ∂Qih ⎟+⎜ ⎟ ⎜ ∂q i − ∂q j ⎠ ⎝

⎞⎤ i ⎟⎥ dq ∧ dq j ∧ dv h + ⎟ ⎠⎦⎥

1 ⎛ ∂Qij ∂Qih ⎜ − 2 ⎜⎝ ∂v h ∂v j

⎞ i ⎟dq ∧ dv j ∧ dv h + ∂Qij dq i ∧ dv j ∧ dt ⎟ ⎠ ∂t . ∂Aij = 0, ∑ h ( i , j ,h ) ∂q Proposition 11.1.5 The differential d t is null if and only if: functions A[ij] are ∂Q jh ∂Qih ∂Qij ∂Qih ∂Qij − = 0, − =0 =0 i j h j ∂ q ∂ q ∂v ∂v independent of vh and t, and ∂t . Proposition 11.1.6 If (M is even and) det (qij)¹0, then the ensemble (Qij, Aij) generates the coefficients of the second kind, to the right, of a non-linear connection to M. Demonstration From the assumption det (qij)¹0, inversing matrix (Qij) , the matrix (Qij) is obtained , whose ∂q h ∂q k ij Q hk = i Q ∂q ∂q j components change, on a local map change, through relationships: . Let’s define the coefficients

R ij = Aih Q hj

, then these coefficients change, on a map change, through the formulas: ∂q n k ∂v i n ∂v i n R h = h δ i + h Ri ∂q k ∂q ∂v ,

which informs that these coefficients are Christoffel’s symbols (to the right) of the second kind, which had to be demonstrated. The calculation is as follows:

258

























⎡ ∂v i ∂q j ⎤ ∂q p ∂q k ∂q i ∂q j Ahp Q pk = ⎢ h Q + A Qmn ij ij ⎥ p m n ∂q h ∂q p ⎣ ∂q ∂q ⎦ ∂q ∂q = k i k i k i i ∂q ∂q ∂q ∂q k ∂v ∂q ∂v ∂q j mn j mn n δ Q Q + δ A Q δ + Aij Q jk m ij m ij i h n h n h n h n ∂q ∂q ∂q ∂q ∂q ∂q = ∂q ∂q , ∂q n k ∂v i n ∂v i n ∂v i ∂q k n ∂q i ∂q k n δ + R R h = h δ i + h Ri i i h n h n k ∂ q ∂ q ∂ q ∂ q ∂ q ∂q ∂v hence: , which means: . e) Bij, Aij and Qij provide the form: 1 1 w = 2 BijdqiÙdqj + AjidviÙdqj + 2 Qij dviÙdvj, This form, based on the condition (11.1.4), gives the space of the speeds an almost symplectic structure distinguished (parameterized by T), changing of its coefficients, on a map change, is made by formulas (11.1.6 '). Rhk =

§ 11.2 POTENTIAL FORM 1 Obtaining a form W , closed, which fulfills Lorentz's conditions, by integrating the system (11.1.2), generally, isn’t an easy operation. We will now seek to present another way of obtaining such forms. 2-closed W form is associated, locally, to a 1-form potential and differential equations (11.2.2), which define it by its coefficients. The potential form implies the writing of Lagrange generalized function (linear in accelerations) and the corresponding Euler-Lagrange equations. The Euler-Lagrange equations, on the manifold of solutions, get the form (11.2.5), equations that, retranscribed, highlight the linear combination (11.2.7), between the equations of the given system and their derivatives. In the general case of a system with an arbitrary number of status parameters, characterized by 2-form W , given by (11.1.7), this being closed, is exact locally and there is therefore a 1-form x , as: x = Vdt + Aidqi + Cidvi,, with the property dx = W. By calculating its differential, this is:

⎛ ∂V ∂Ai ⎞ i ⎛ ∂A ⎜⎜ i − ⎟⎟dq ∧ dt + 1 ⎜ j − ∂Ai ∂t ⎠ 2 ⎜⎝ ∂q i ∂q j ∂q dx = ⎝

⎞ i ⎟dq ∧ dq j + ⎟ ⎠

1 ⎛ ∂C j ∂C i ⎞ i ⎟dv ∧ dt + ⎜⎜ i − j 2 ⎝ ∂v ∂v ⎠ By identifying its coefficients with W’s, this system is obtained: ⎛ ∂C j ∂A + ⎜⎜ i − ij ∂v ⎝ ∂q

⎞ i ∂V ∂C i ⎟dq ∧ dv j ⎛⎜ − ⎟ i ∂t ∂ v ⎠ ⎝ +

∂Ai ∂V ∂C i ∂V − i = − Ei , − i = − Pi , ∂t ∂q ∂t ∂v ∂A j ∂Ai ∂C j ∂Ai − j = Bij , − = − Ai j , i ∂q ∂q ∂q i ∂v j ∂C j ∂C i − = Qij . ∂v i ∂v j 11.2.1

⎞ i ⎟dv ∧ dv j ⎟ ⎠

If operators are considered (linked to the local map): ∂A ∂C h h ∂ξ ∂ ∂V :ξ → =− dt + h dq h + dv , ∂z ∂z ∂z ∂z ∂z where z = (qi, vi), Lorentz's conditions can be transcribing, based on relationships (11.2.1), in the form of: ∂ξ ∂ξ ( S ), ( S ), i i (11.2.1') S(Ai) = ∂q S(Ci) = ∂v ∂ ∂ ∂ S = + vh h + F h h , ∂t ∂q ∂v is the derivation operator along the trajectories of the system. where A solution of the system (11.2.1') provides a potential x for W.

259





















⎛ ∂Φ ⎛ ∂Φ ⎞ ξ = ξ + dΦ = ⎜ + V ⎟dt + ⎜⎜ Ai + i ∂q ⎝ ∂t ⎠ ⎝

⎞ i ⎛ ∂Φ ⎞ ⎟⎟dq + ⎜ C i + i ⎟dv i ∂v ⎠ ⎝ ⎠ , ∂Φ + V = 0, doesn't change the field. With each potential you can choose the F function so that ∂t with which the change of potential is performed, then the potential is simplified. The form x becomes: x = Aidqi + Cand you, so that its differential is: ⎛ ∂C j ∂Ai ⎞ i ∂A 1 ⎛ ∂A j ∂Ai ⎞ i ⎜ ⎟dq ∧ dq j + ⎜ − − j ⎟⎟dq ∧ dv j − i dq i ∧ dt + i j ⎟ i ⎜ ⎜ 2 ∂q ∂t ∂q ⎠ ∂v ⎠ ⎝ ∂q dx = ⎝ −

∂C i 1 ⎛ ∂C j ∂C dv i ∧ dt + ⎜⎜ i − ji ∂t 2 ⎝ ∂v ∂v

⎞ i ⎟dv ∧ dv j ⎟ ⎠ .

In this case, the equations (11.2.1) become: ∂C j ∂Ai ∂A j ∂A ∂Ai ∂C i Aij = − j Bij = i − ij Qij = ∂C j − ∂Ci i i j ∂q ∂v , ∂q ∂q , ∂v ∂v , Ei = - ∂t ,Pi = - ∂t . Considering Lorentz’s conditions too, that is (11.1.9) and eliminating from all these equations the functions Aij, Bij, Qij, Ei and Pi, we have: ∂A j ∂C j j S ( Ai ) = i v j + F , ∂q ∂q i ∂A j ∂C j S( Ci ) = i v j + i F j . ∂v ∂v 11.2.2 The system (11.2.2), as well as (11.2.1 '), has a solution, based on the theorem of existence and ∂Ai ∂C i ∂A ∂C , uniqueness, as we can express the derivatives ∂t =F(t, Q, V, A, C, ∂z ∂z ) =-Ei and ∂t =G(t, Q, V, A, ∂A ∂C , C, ∂z ∂z ) =-Pi. This system defines the coefficients of the potential form. In relation to a local map change, the form (11.1.11), the 1-shape x coefficients change according to the formulas: ∂q i C h = h Ci , ∂q ∂v i ∂v i Ah = h C i + h Ai , ∂q ∂v (11.2.3.) which informs that z = Cidqi is a d-form per M, while ξ is a d-form on TM. Conclusions Reunited, formalities we presented above are called Gallissot-Souriau's formalism (GS). As has been stated, it is called direct problem in the formalism G-S giving to a 2-form presymplectic W on R´TM (of rank 2m and closed) and obtaining a spray (force field), for which W is a Lagrange form. Its resolution consists in writing characteristic equations and putting them into Newtonian equivalent form. Basically, is given, in the most general case, a 2-form (11.1.7), closed and of 2m rank. In the hypothesis (11.1.10), from equations (11.1.9) is expressed Fi, and then the equations of Newton (11.1.1) shall be written. It is called inverse problem in the formalism G-S expressing equations (11.1.1) and determining a 2form W, presymplectic on R´TM (closed and of rank 2m), whose characteristic are exactly the solutions of Newton's equations. The result is that any Newtonian system (nonndegenerate) accepts a G-S representation, the proposition 1.2.2 '. This is a particular case of the 1.2.2 theorem. 2 Lagrange's function. Euler-Lagrange's equations

260













The x potential is determined up to a differential. A potential change in form:

∂ ∂ ∂ ∂ d + vi i + ai i + bi i . ∂q ∂v ∂a where dt operator is: ∂t An immediate calculation leads to: ∂L ∂C h h ∂Ah h ∂L ∂L ∂C h h ∂Ah h = a + i v , = Ci , = i a + i v + Ai , i i ∂q i ∂q i ∂q ∂a ∂v ∂v ∂v d ⎛ ∂L ⎜ dt ⎝ ∂a i

∂C ∂C ⎞ dC i ∂C i = + v h hi + a h hi , ⎟= ∂t ∂q ∂v ⎠ dt

2 ∂ 2 Ci h ∂ 2 Ci h ∂ 2 Ci h k d 2 ⎛ ∂L ⎞ d 2 C i ∂ C i + 2 v + 2 a + v v + = = ⎜ ⎟ ∂t 2 ∂t∂q h ∂t∂v h ∂q h ∂q k dt 2 ⎝ ∂a i ⎠ dt 2

+2

∂ 2 Ci h k ∂ 2 C i h k ∂C i h ∂C i h a v + a a + ha + hb ∂v h ∂q k ∂v h ∂v k ∂q ∂v .

Matriceally, this last relationship can be put in the form of:

⎛ ∂ 2 Ci ⎜ 2 ⎜ ∂t ⎜ ∂ 2 Ci d 2 ⎛ ∂L ⎞ h h = ( 1 , v , a ) ⎜ h ⎜ ⎟ dt 2 ⎝ ∂a i ⎠ ⎜ ∂q ∂t ⎜ ∂ 2 Ci ⎜ h ⎝ ∂v ∂t

∂ 2 Ci ∂t∂q k ∂ 2 Ci ∂q h ∂q k ∂ 2 Ci ∂v h ∂q k

∂ 2 Ci ∂t∂v k ∂ 2 Ci ∂q h ∂v k ∂ 2 Ci ∂v h ∂v k

⎞ ⎟ ⎟⎛ 1 ⎞ ⎟⎜ k ⎟ ⎟⎜ v ⎟ ⎟⎜ a k ⎟ ⎟⎝ ⎠ ∂C ∂C a h hi + b h hi ⎟ ∂v . ⎠ + ∂q

Next, we have: 2 ∂A ∂A ∂A d ⎛ ∂L ⎞ ∂Ai ∂ 2Ch h ∂ Ah + v h hi + a h hi + a h hi a h ⎜ i ⎟= + v + dt ⎝ ∂v ⎠ ∂t ∂q ∂v ∂v + ∂t∂v i ∂t∂v i

a hv k

2 2 2 ∂ 2Ch h k ∂ Ak h k ∂ Ah h k ∂ Ch h ∂C h + a v + v v + a a + b ∂q k ∂v i ∂v h ∂v i ∂q k ∂v i ∂v k ∂v i ∂v i Regrouped, these

relationships lead to the equations: Ei =

∂ 2 Ci ∂ 2 Ci h ∂ 2 Ci h ∂ 2 Ci h k − 2 v − 2 a − v v − 2 h h h k ∂ t ∂ t ∂ q ∂ t ∂ v ∂ q ∂ q -

−2 +

∂ 2 Ci h k ∂ 2 C i h k ∂C i h ∂C i h a v − a a − ha − hb ∂v h ∂q k ∂v h ∂v k ∂q ∂v +

2 2 ∂Ai ∂A ∂A ∂A + v h hi + a h hi + a h hi a h ∂ C h + v h ∂ Ah + ∂t ∂q ∂v ∂v + ∂t∂v i ∂t∂v i

2 2 2 ∂ 2Ch ∂C h ∂C h h ∂Ah h h k ∂ Ak h k ∂ Ah h k ∂ Ch − i a − i v . a v +a v +v v +a a + bh k i h i k i k i ∂q ∂q ∂q ∂v ∂v ∂v ∂q ∂v ∂v ∂v ∂v i h

k

These relationships, reorganized, become: ⎛ ∂Ai ∂Ah ⎞ ⎛ ∂C h ∂C i ⎞ ⎜ − i ⎟⎟ ⎜ i − h⎟ h ∂q ⎠ h ∂v ⎠ bh + ⎜⎝ ∂q ⎝ ∂v v +

⎛ ∂Ai ∂C h ⎜ ⎜ ∂v h − ∂q i ⎝

⎞ ⎛ ∂Ah ∂C i ⎟ ⎜ ⎟ ⎜ i − ∂q h ⎠ ah + ⎝ ∂v

261



















The dynamic system (11.1.1) was associated to 2-Lagrange form (11.1.7), with fulfilling Maxwell's principle (11.2.10). Locally, it is associated to 1-potential x form . By definition, we call Lagrange function, the function given on J2M by: (11.2.4) L = Ci + Aivi. The function (11.2.4) corresponds to Euler-Lagrange equations, deducted on J3M by: d 2 ⎛ ∂L ⎞ d ⎛ ∂L ⎞ ∂L − 2 ⎜ i ⎟ + ⎜ i ⎟ − i = 0, (11.2.5) Ei = dt ⎝ ∂a ⎠ dt ⎝ ∂v ⎠ ∂q

⎞ ∂Ai ⎟ ⎟ ⎠ ah + ∂t +



















d ⎛ ∂C i ⎜ dt - ⎝ ∂t

⎛ ∂Ah ∂C i ⎜ ⎜ ∂v i − ∂q h ⎝

⎛ ∂Ah ∂C i ⎜ ⎜ ∂v i − ∂q h ⎝

∂ ⎛ ∂C h ∂C i ⎞ ∂ ⎜ i − i ⎟ k ∂v ⎠ ∂t a H + V H + ∂q v h v k + ⎝ ∂v ⎛ ∂C h ∂C i ⎞ ∂ ⎛ ∂C h ∂C i ⎞ ∂ ∂ ⎛⎜ ∂Ah − ∂C i ⎞⎟ ⎜ i − h⎟ ⎜ i − h⎟ i k k ∂q h ⎟⎠ ∂v ⎠ ahvk + ⎝ ∂v ∂v ⎠ v h a K + ∂v a h a K. ∂v k ⎜⎝ ∂v + ∂q ⎝ ∂v Taking into account the relationships between the coefficients of Ω and those of dξ, we obtain Euler – Lagrange equations: ⎞ ∂ ⎟ ⎠ + ∂t

⎞ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎠

⎛ ⎛ d da j ⎞ dv j ⎞ ⎟⎟ − Aij ⎜⎜ a j − ⎟ Pi − A ji v j − Qij a j − E i + Bij v j − Aij a j + Qij ⎜⎜ b j − dt dt ⎠ dt ⎟⎠ ⎝ ⎝ Ei= = 0. A solution of these

(

)(

)

vi =

dv i d 2 q i dq i i = , a = dt dt 2 and dt

equations is a curve on M, qi = qi(t), which raised to J3M through da i d 2 v i d 3 q i bi = = = dt dt 2 dt 3 , fulfills the same relationships above and therefore we have the respectively equations:

d Pi − A ji v j − Qij a j − E i + Bij v j − Aij a j dt Ei= = 0.

(

)(

)

(11.2.6) In particular, among the solutions of the system (11.2.6) we have all the system solutions (curves for which):

Ei + Bij v j − Aij a j = 0, P − A v j + Q a j = Φ (t , q, v, a) = cons. / trajectories,

ji ij i 11.2.7 i where F i are conservation laws (prime integrals to the system).

qi − F i ( t , q , q ) = 0 equations define in J2M the manifold of solutions: Ker( qi − F i ( t , q , q )) . On

this manifold the equations (11.2.6) become:

Ei + Bij v j − Aij F j = 0 , Pi − A ji v j − Qij F j = 0 , which are nothing but Lorentz's conditions.

Ei + Bij v j − Aij a j

Equation structure (11.2.6) requires the study of functions Fi = j

Pi − A ji v + Qij a

and Pi

=

j

, which can be called superforce and superimpulse, and that fulfill the relationship: dP i / dt =Fi.

Another interpretation Euler – Lagrange's equations (11.2.5), on the manifold of solutions , can be written in the form:

[

qi - F i = 0

]

d Qij (q j - F j ) - Aij (q j - F j ) = 0 , dt Ei =

which informs that they are linear combinations between the equations of the given system and their derivatives, result that was highlighted in [26]:

Qij (q j - F j ) • +

dQij dt

- Aij

(q j - F j )= 0

(11.2.8) Ei = These equations are written in the main form:

.

Qij q j + Ri = 0 ,

262























Ri = −Qij where

⎞ dF j ⎛ dQij + ⎜⎜ − Aij ⎟⎟ q j − F dt ⎝ dt ⎠

(

j

)

. The system (11.2.21) is self-adjoint and the matrix rank (Qij) i i indicates the degree of complexity of the system q − F = 0 . More generally, given the coefficients Aij, Bij, Qij and Ei, the coefficients of Pi can be obtained by:

Pi

Traiectorii

= A ji v j + Qij F j + ∫

Traiectorii

( E i + Bij v j − Aij F j )dt

or:

Pi

Traiectorii

,

= A ji v j + Qij F j .

We have: The theorem 8.5.1 The equations Euler – Lagrange (11.2.5), associated with L (11.2.4) function, accept, among their solutions, the trajectories of the dynamic system (11.1.1). The L function is a Lagrange function for the system (11.1.1). Indeed, considering ai = Fi and taking into account Lorentz's conditions (11.1.9), the equations (11.2.5) are identically fulfilled. From the presentation made in this paragraph, as in the entire third chapter, it follows that the inverse issue occurs: if a system of Newtonian equations is given, they are assigned a superfield, through a 2-form W type (11.1.7), defined on J1M and closed. Reciprocally, there is also the direct problem: if a closed 2-form W is given , it implies, through the formulas (11.2.6), the existence of functions Fi and Pi and these, cancelled, define the spray S which implies, through its coefficients, the Newtonian system. In conclusion, the description of the evolution of a Newtonian dynamic system, through equations of type (11.1.1), and the existence of an associated superfield are equivalent. In other word, these facts describe two sides of the same reality.

§ 11.3 SUPERFIELDS AND SUPERWAVES ASSOCIATED TO NEWTONIAN DYNAMIC SYSTEMS. FIELD THEORY In this paragraph it is shown that any dynamic Newtonian system is canonically associated with a superfield and superwaves of the latter’s propagation, defined on the space of the first order jets: J1M. The associated superfield is unique in the original given conditions. As we know, in the Newtonian formalism of classical mechanics is given, for a mechanical system, a i  field of forces F = f (t, q, q ) (dynamic system) and, based on Newton's equations: q = Fi, (m= 1), we can follow the evolution of the system. 11.3.1 Lagrange form Given M a differentiable manifold, class C ∞ and of finite dimension m, in which a dynamic system S is defined . Its evolution is described by a system of Newtonian differential equations as (11.1.0), where the field of forces F is presumed differentiable. Newton's motion equations, expressed in the space of evolution (space of the states) J1M =R´TM, can be written in the form:

dv i − F i dt = 0 , i i 11.3.1 dq − v dt = 0.

The evolution of such a system is defined with the mention of the force field F=F(t,q,v). i

Being given a dynamic Newtonian system (written in dynamic form v =F(t,q,v), which we generally assume classical non-Lagrangean. This system can be written in equivalent kinematic form (11.3.1) Gallissot associated to the system (11.3.1) 2-form, which we call Lagrange-Gallissot form:

Ω G = dv i ∧ dq i + ( F i dq i − v i dv i ) ∧ dt . It was generalized by Souriau for some Lagrangian dynamic systems by: 11.3.2

Ω S = dv i ∧ dq i + ( E i dq i − v i dv i ) ∧ dt +

for which Lorentz's condition is satisfied:

1 Bij dq i ∧ dq j , 2

Ei = Fi − Bij v j , 263

(

Fi = Ei + Bij v j

)























Ω L = A ji dv i ∧ dq j + ( E i dq i − Pi dv i ) ∧ dt +

1 1 Bij dq i ∧ dq j + Qij dv i ∧ dv j . 2 2

11.3.3 We have: Proposition 11.3.1 Characteristic curves of 2-form (11.3.3), given by equations:

Bij dq j − Aij dv j + Ei dt = 0 , A ji dq j + Qij dv j − Pi dt = 0 ,

11.3.4 are exactly the trajectories of the system (0.1), if and only if Lorentz's conditions are met :

⎧⎪ Fi = Ei + Bij v j − Aij F j = 0 , ⎨ j j ⎪P = Pi − A ji v − Qij F = 0 , 11.3.5 ⎩ i

we note with:

:

⎛ Bij det ⎜⎜ ⎝ A ji (11.3.6) det Δ =

− Aij ⎞ ⎟≠0 Qij ⎟⎠

The conditions (11.3.5) delimit, in part, the arbiter of Wform coefficients . Let's call, through abuse of language, solution of the system (11.3.1) a 2-form W , like (11.3.3), fulfilling relationships (11.3.5) and we represent it through our own Δ matrix . Observation While 2-form (11.3.3) is defined by five groups of coefficients: (Aij, bij,Qij, Ei, Pi), one of its solutions is defined only by three such groups (Aij, Bij, Qij) and describes the dynamics of (11.3.1). However, it has dynamic, not geometric character. Proposition 11.3.2 Any Newtonian system (nondegenerate) accepts a Gallissot-Souriau representation. Proposition 11.3.2 is a particular case of 1.2.2 theorem. Indeed, we can associate to the system of Newtonian equations a 2-form of type W G. In its equivalence class there is then one that is (at least) closed. Observation In general we cannot say a priori, if the det (bij) is different from zero or not. If the j j ~j system (11.3. 8) accepts two solutions (g i ) and ( g i ), then both their sum and (lg i ) are also solutions, " l Î j ~j R. det ( g i +lg i ) is cancelled only for a finite number of values l , for any other value it is therefore different from zero.

§ 11.3.2 Maxwell's principle As in any class of equivalent forms, there are closed forms. Such forms describe very well the evolution of dynamic systems in the classical theory of the field, as well as many other cases of dynamic systems. These requirements, ranked as principle [20], let us achieve new and interesting results. On the basis of 1.2.2 theorem (proposition 11.3.2), we deduce that being given a dynamic system (11.3.1) (through the field of forces Fi), the form of Lagrange (11.3.3) that can be associated with fulfilling the relationships (11.3.4), is not unique, their set can diminish, imposing the condition (called Maxwell's principle) to be closed (d W = 0). This condition requires that its coefficients constitute a solution to the system of partial derivative equations, called Maxwell's equations:

∂Bij

∂Qij ∂Ahj ∂Ahi ∂Aih ∂A jh − + i − j = j i ∂v ∂q ∂q = 0, ∂q h ∂v ∂v 0, ∂Aij ∂Pj ∂Ei ∂Bij ∂E j ∂E i + i − = + i + j = j ∂t ∂t ∂q ∂v ∂ q ∂ q 0, 0, h

(11.3.7)

∂Qij ∂t

+

−

∂Pj ∂v i

+

∂Bij ∂Qij ∂Pi = ∑ = ∑ = h h j ∂v 0, ( i , j ,h ) ∂q 0, ( i , j ,h ) ∂v 0.

where the sum is obtained after the circular permutations of indices (i, j, k) in the last two relationships (1£ i,j,k£2m). The equations of the system (11.3.7) generalize Maxwell's classical equations in electrodynamics, if the dynamic system describes the evolution of charged masses, placed in an external electromagnetic field.

264













A first geometric object, associated with a dynamic system, is Lagrange form. A dynamic Newtonian system, given through equations (0.1), can be associated to a ''2-form Lagrange'' complete [17]:





A 2-form W , associated with the dynamic system (11.1.0), which fulfills Maxwell's principle, is called superfield, its coefficients are called the superfield coefficients and equations (11.3.7), superfield equations. Therefore, any dynamic system is associated with a field. This field is not unique. The coefficients Aij, Bij, Qij, Ei, Pi, which satisfy the condition (11.3.3), Lorentz's conditions (11.3.4) and Maxwell's equations (11.3.7), are called superfield coefficients, the form W is called superfield form. Given a solution of the equations (11.3.3), any other system of functions constitutes a solution if and only if they differ from the solution given by a first integral. Under initial conditions, the equation solution (11.3.3) exists and is unique [1]. 11.3.3 An integrant factor:

(Cαβ )which transforms the system (11.3.1) into the self-adjoint system: Bij q j − Aij v j − ( Bij v j − Aij F j ) = 0 , A ji q j + Qij v j + A ji v j − Qij F j = 0 ,

and that highlights the property that the system trajectories are exactly the 2-form Wcharacteristics , leads to: Theorem 11.3.3 The notion of superfield associated with a dynamic Newtonian system, via the form of Lagrange, fulfilling Lorentz's conditions and closed, is identical to the notion of the integrant factor of the kinematic system associated with the dynamic system given.  Demonstration Being given the dynamic system q = F(t,q, q ), and implicitly the associated kinematic system (11.3.1). The conditions of the self-adjunction are, by the definition of the integrant factor:

Bij

(C )= ⎛⎜⎜ A αβ

⎝

ji

− Aij ⎞ ⎟ Qij ⎟⎠

, fulfilled. These conditions of the self-adjunction translated, with the use of Lorentz's conditions, by derivation, lead to the equations of Maxwell (11.3.7). ∂ ∂ S = vi i + F i i ∂q ∂v , we have Lorentz’s conditions 11.3.4 Being given an autonomous spray S on TM, (11.3.5). There we deduce that the functions Ei and Pi can be considered as components of a form q = Eidqi+ Pidvi, obtained from the spray as the covariant expression of the spray by the almost symplectic form:

ω = A ji dv i ∧ dq j +

1 1 Bij dq i ∧ dq j + Qij dv i ∧ dv j 2 2 ,

with which W L = w Ù + q dt. The geometrical interpretation of the function shall be immediately deducted: q(S) = Eivi - PiFi.

11.3.5 Field theory (adapted to the study of dynamic systems of second order)

The considerations set out in paragraph 0.4 shall be adapted to the dynamic systems of the second order as follows. 1 Canonical isomorphism Let’s consider the manifold TM and on it the w autonomous form. At any point (q, v)ÎTM, w defines an isomorphism, which we note w (q, v), of the tangent space T(q,v)(TM) on the cotangent space T*(q,v)(TM), as follows. Given X, YÎT(q, v)TM, then the application: ω (Xq ,v )

(Y) =

ω (Xq ,v )

( X ,Y ) is linear, so it's a covector from

T(*q ,v )TM

ω (Xq ,v ) ,

defined by

.

ωX If XÎX(TM) is a field of vectors on TM, then the application w x:(Q, v) ® ( q ,v ) , depends differentiably on (q, v)ÎTM and as such is a 1-form, ie w xÎL1(TM). From the property of the 2-form w of having the determinant of its coefficients different from zero, it X ( q ,v ) ∈ T( q ,v )TM → Λ1( q ,v )TM follows that correspondence is an isomorphism. Therefore, whatever l Î L ω Xλ = λ 1TM, there is an XlÎX(TM) field so that through it we have: .

265

























Of the solutions of the system (11.3.7) we choose those that meet the conditions Bij = – Bji , Qij = – Qji and the functions E i and Pi are given by (11.3.5).































⎛ ∂ ∂ ⎞ ⎜ ⎟ ⎜ ∂q i , ∂v i ⎟ ⎝ ⎠ , the w form components are: Given a base (on the geometrical area of a vector map) ⎛ ∂ ⎛ ∂ ⎛ ∂ ∂ ⎞ ∂ ⎞ ∂ ⎞ ∂ ⎞ ⎛ ∂ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ i , j⎟ ⎜ ∂q i , ∂q j ⎟ ⎜ ∂q i , ∂v j ⎟ ⎜ ∂v i , ∂q j ⎟ ⎠ =Bij ,w ⎝ ⎠ =-Aij ,w ⎝ ⎠ =Aji ,w ⎝ ∂v ∂v ⎠ =Qij . w⎝ From the above it follows that the form w defines on the manifold (orientable) TM an almost symplectic structure. In the case of Newtonian dynamic systems the situation is presented in the following way. Being given the dynamic system (11.1.1), it is associated to the 2-form of Lagrange (11.1.7). It can now be defined a natural isomorphism, noted with Δˆ , T (R´tm) ® T*(R´tm), expressed locally by:

with:

⎛ Bij ⎜ ⎜ A ji ⎜− E j (Dab) = ⎝

− Aij Qij Pj

⎛ Bij Ei ⎞ ⎛ a ⎞ ⎛Xi⎞ ⎟ ⎜ i⎟ ⎜ ⎜ ⎟ i − Pi ⎟ ⎜ bi ⎟ ⎜ Y ⎟ → ⎜ A ji ⎜⎜ ⎟⎟ ⎜− E Z ⎠ 1 ⎟⎠ ⎜⎝ c ⎟⎠ 1,2 m + 1 j ⎝ ⎝ : =, (A, B =),

− Aij Qij Pj

Ei ⎞ ⎛ X j ⎞ ⎟⎜ ⎟ − Pi ⎟ ⎜ Y j ⎟ ⎜ ⎟ 1 ⎟⎠ ⎜⎝ Z ⎟⎠

ai =BijX j - AijY j+EiZ, bi = AjiX j + QijY j+PiZ, c = -E iXi+PiYi+ Z. Proposition 11.5.1 The condition necessary and sufficient for the non-autonomous semi-spray

∂ ∂ ∂ + vi i + F i i ∂t ∂q ∂v to be taken, by canonical isomorphism, to dT, ( Δˆ (S) = dt), is to meet Lorentz's S= conditions (and, as a consequence, we have the law (11.1.8 ')). Indeed, from:

j

j

⎛ Bij − Aij Ei ⎞ ⎛ v j ⎞ ⎛ Bij v − Aij F + Ei ⎞ ⎛ 0 ⎞ ⎛ vi ⎞ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎜ ⎟ Qij − Pi ⎟ ⎜ F j ⎟ ⎜ A ji v j + Qij F j − Pi ⎟ ⎜ 0 ⎟ ⎜ F i ⎟ → ⎜ A ji ⎜⎜ ⎟⎟ ⎜ ⎟ ⎜− E 1⎠ Pj 1 ⎟⎠ ⎜⎝ 1 ⎟⎠ ⎜⎝ − E j v j + Pj F j + 1 ⎟⎠ ⎜⎝ 1 ⎟⎠ ˆ j ⎝ ⎝ Δ : = = , this statement follows. 2 Volume form on TM Taking into account the formulas of changing coefficients of w , on a change of map, we deduce: 2

4

⎡ D( q , v ) ⎤ ⎡ D( q ) ⎤ det Δ = ⎢ det Δ = ⎢ ⎥ det Δ ⎥ det Δ D( q ) ⎦ ⎣ D( q , v ) ⎦ ⎣ , that is . D( q ) Noting J (q) = D( q ) , we have the relationship: (11.3.8)

det Δ =

1 J2

det Δ

.

1 m On the other hand, between the differential forms dq1Ù...ÙdqmÙdv1Ù...Ùdvm şi dq ∧ ... ∧ dq ∧

∧ dv 1 ∧ ... ∧ dv m ,defined on UiÇUj, intersection, of two geometric areas, we have the relationship: 1 m 1 m (11.3.9) dq ∧ ... ∧ dq ∧ dv ∧ ... ∧ dv = J2(q) DQ1Ù. .. Ù dqmÙdv1Ù. .. Ù youm. Dividing part with part relationships (11.3.1) and (11.3.2), we get the formula: 1

m

1

m

det Δ dq ∧ ... ∧ dq ∧ dv ∧ ... ∧ dv = det Δ dq1Ù...ÙdqmÙdv1Ù...Ùdvm, formula that expresses the law of change, defined by it, by switching from one base to another. Rank 2m differential form: (11.3.10) h = det Δ dq1Ù...ÙdqmÙdv1Ù...Ùdvm, is a local volume form on TM, and the law of change extends to the whole field on which the dynamic system is defined. The local volume form is invariant (does not change the sign), no matter if the map change on M keeps the orientation or not.

266

















Given the volume form h, on the almost symplectic manifold (TM,w), it defines, with the help of the inner product, an isomorphism of the tangent T TM) fibrate on the vector fibrate L2m-1(TM) of the forms (11.3.

m-1)-linear alternant defined on TM and with real values.

i η To a tangent vector X(q, v)ÎT(q, v)TM we associate (11.3. m-1)-form X ( q ,v ) . Given the form z Î L 2m-1(TM), arbitrary and (U,c) a local map. Then z has the expression: m

1 i m 1 m ∑ α i dq ∧ ... ∧ dqˆ ∧ ... ∧ dq ∧ dv ∧ ... ∧ dv

z =

i =1

+

m

1 m 1 j m ∑ β j dq ∧ ... ∧ dq ∧ dv ∧ ... ∧ dvˆ ∧ ... ∧ dv

+ j =1

Let’s calculate the X field so that ixh= z , then from: m

IX h = m

∑ ( −1 )

i −1

det Δ X i dq 1 ∧ ... ∧ dqˆ i ∧ ... ∧ dq m ∧ dv 1 ∧ ... ∧ dv m

i =1

∑ ( −1 )

m + j −1

+ j =1 it follows, by identifying with z :

det ΔY j dq 1 ∧ ... ∧ dq m ∧ dv 1 ∧ ... ∧ dvˆ j ∧ ... ∧ dv m

X i = ( −1 )i −1

αi

Y i ( −1 ) m +i −1

+

,

βi

det Δ , det Δ . This result confirms the property that the application considered is, indeed, an isomorphism. If X is a field of vectors on TM, through this isomorphism we obtain a (n= 2m-1)-differential form. 11.3.6 Differential operators on an almost symplectic manifold (N = TM,w) 1a Adjunction operator of Hodge-de Rham Definition 7.1 Given, in general, a differentiable manifold of dimension N, endowed with a two-fold covariant tensor, nondegenerate w , h, the volume form associated to w and L (N) external algebra defined on N. We call Hodge-de-Rham's adjunction operator and we note it with ∗ , the application ∗ : L ® L by which each p-forms a Î L p(N) is associated to (n-p)-form ∗ aÎLn-p(N) by formula: ~ ~ ~ ( ∗ a)(X1,...,Xn-p) η = aÙ Δ X1Ù...Ù Δ Xn-p. Locally, this operator shall be expressed by: ( ∗ a)

α

i1 ,...,i p1

i p +1 ...in

~ ~i j = Δ i1 j1 ...Δ p p α j1 ,..., j p

where bijection between L p and Ln-p .

= (1/p!)

i1 ,...,i p ~ η , i1 ,...,i p ,i p +1 ,...,i n α

are the "contravariant" components of p-form a . ∗ operator is a

We have the following important relationship:

∗ 2 =ew(-1)p(n-p)id Λ( M ) , hence, together with the property of bijectivity, its inverse: ∗ -1 =ew(-1)p(n-p) ∗ , w sign and where e w = (-1) , ∗ (wX) is a (n-1)-form. 2a Co-differentiation operator Definition 11.3.5 Being given N, an almost symplectic space. It is called co-differentiation operator on N application d :L(n) ® L (n), defined on L p(N) with values in Lp-1(N) by formula: d = (-1)p+1 ∗ -1d ∗ . It is immediately inferred that we have the property d 2 = 0, as well as d f = 0, if fÎL° (M). 3a Laplace operator Definition of 11.3.6 We call Laplace-d'Alembert operator and it is noted with on a presymplectic manifold, the application whereby each p-form is associated with a p-form through the formula: =dd+dd : a®a . A p-form w is called harmonic form if w = 0. 1 α = a ij dx i ∧ dx j , 2 If in a local 2-form map a is expressed by then

267

⎛ Bij ~ ⎜ Δ = ⎜ A ji ⎜− E j ⎝

− Aij

⎛ ai ⎞ ⎛ Bij ⎜ ⎟ ⎜ ⎜ bi ⎟ = ⎜ − A ji ⎜λ⎟ ⎜ 0 ⎝ ⎠ ⎝

Aij Qij

⎛ X i ⎞ ⎛ Q ij ⎜ ⎟ ⎜ ⎜ Y i ⎟ = ⎜ A ji ⎜⎜ ⎟⎟ ⎜⎜ ⎝ λ ⎠ ⎝ 0

− Aij B ij 0

Qij Pj

Ei ⎞ ⎟ − Pi ⎟ 1 ⎟⎠

, ~ ~ = η ∧ dt η η η As volume form on R´TM we extend form through . * The isomorphism w : T(TM)®T (TM) can be extended to an isomorphism ∂ ∂ ∂ Xi i +Yi i +λ ∂t → a i dq i + bi dv i + λdt , ∂q ∂v W:X(R´TM)®L1(R´TM): expressed by:

0

with the inverse:

0 ⎞⎛ X j ⎞ ⎟ ⎟⎜ 0 ⎟⎜ Y j ⎟ 1 ⎟⎠⎜⎝ λ ⎟⎠ ,

0 ⎞⎛ a j ⎞ ⎟⎜ ⎟ 0 ⎟⎜ b j ⎟ ⎟ 1 ⎟⎠⎜⎝ λ ⎟⎠

.

11.3.7 Field theory Being given the differentiable manifold of R´TM, (to which a double structure ~ ( Ω , Δ ) is also assigned) and on it a 2-closed form a ; we will associate to it, as follows, a field theory (see subsection 0.4). We present these results in the table:

Nature

Potential

Field

Current

Forms of definition:

xÎLm-1(M), l=*xÎL1(M),

aÎL2(M), b=*aÎLm-2

gÎLm-1(M), q=*gÎL1(M),

Maxwell type equations (field):

dx=0, dl-a=0,

da=0, db+g=0,

dg=0, dq-n=0,

Propagation equations (waves):

ÿx=ddx,

ÿa=dda,

ÿg=ddg.

Maxwell type equations Dual forms (field)

dl=0, dx+b=0,

db=0, da-q=0,

dq=0, dg+n=0,

Propagation equations Dual forms (waves)

ÿl=ddl,

ÿb=ddb,

ÿq=ddq.

As discussed above, each dynamic system, defined on a manifold M, is associated with a closed W 2form , defined on the almost symplectic R´TM manifold. It plays a 2-form a role and therefore is associated with a field theory and, consequently, a field and waves, as solutions of the propagation equations. 11.3.8 Superfields associated with general Newtonian dynamic systems

268

















1 α= ( i j 2 a i j )dx ∧ dx . Obviously, these operators are defined on the manifold V = TM. 4o Extensions of differential operators to R´TM Let's consider the nonautonomous case now. For the construction of a field theory, it is necessary to have a two-fold covariant tensor, nondegenerate, on the manifold R´TM linked to the given dynamic system. Such a tensor, associated intrinsically to the system, is:













q i = q i ( q h ), v i =

∂q i h ∂q i ∂q i h ∂v i i h i v , d q = dq , d v = dv + h dq h , h h h ∂q ∂q ∂q ∂q

(11.3.11) coefficients of W form are subject to the exchange relationship:

⎛ Bhk ⎜ ⎜ Akh ⎜− E k ⎝

− Ahk Qhk Pk

relationship that implies: Ph =

Ahk = (11.3.12)

⎛ ∂q i ⎜ ⎜ ∂q h ⎜ Eh ⎞ ⎜ 0 ⎟ ⎜ − Ph ⎟ ⎜ 0 ⎜ 0 ⎟⎠ ⎜⎝ =

∂q i Pi , ∂q h

Eh =

∂v i ∂q h ∂v i ∂v h 0

⎞ 0⎟ ⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟⎟ ⎠

⎛ Bij ⎜ ⎜ A ji ⎜− E j ⎝

∂q i ∂v i E − Pi , i ∂q h ∂q h

− Aij Qij Pj

Qhk =

∂q i ∂v j 1 ⎛ ∂q i ∂v j ∂q j ∂v i ⎜ A − − ij 2 ⎜⎝ ∂q h ∂q k ∂q h ∂q k ∂q h ∂q k

Bhk =

Ei ⎞ ⎟ − Pi ⎟ 0 ⎟⎠

⎛ ∂q j ⎜ ⎜ ∂q k ⎜ ∂v j ⎜ k ⎜ ∂q ⎜ 0 ⎜⎜ ⎝

0 ∂v j ∂v k 0

⎞ 0⎟ ⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟⎟ ⎠,

∂q i ∂q j Qij , ∂q h ∂q k

⎞ ⎟⎟Qij , ⎠

∂q i ∂q j ∂v i ∂q j ∂v i ∂q j 1 ⎛ ∂v i ∂v j ∂v j ∂v i ⎜ B − ( − ) A + − ij ij 2 ⎜⎝ ∂q h ∂q k ∂q h ∂q k ∂q h ∂q k ∂q h ∂q k ∂q k ∂q h

⎞ ⎟⎟Qij , ⎠

Interpretations Functions Pi are components of a 1-d-form (impulse) on M, functions (Ei, Pi) are the components of a 1-d-form on TM, 2 Maxwell's principle We will say that a 2-form as (11.1.7), with condition verification (11.1.10), fulfills Maxwell's principle [3] if closed. By cancelling outer differentiation, it follows that Maxwell's equations are satisfied (11.3.7) 11.3.9 Superfield equations. Wave equation 1a Volume element On R´TM we build the following volume element:

η=

det Δ dq 1 ∧ ... ∧ dq m ∧ dv 1 ... ∧ dv m ∧ dt

with D matrix defined by (11.1.10). 2a Differential operators: Hodge – de Rham’s adjunction operator, the codifferentiation operator, and Laplace– d'Alembert operator are defined as in the classical case. 3o Equation of superfield propagation (superwaves equation is defined in the same manner). If W is 2-form of Lagrange (closed), superfield propagation equation is: W = ddW . Observations From the transformation formulas (11.3.12), of 2-form W components, it is apparent, on the one hand, that the functions Qij and Pi constitute the distinguished objects: P = Pidqi (a covector – 1 impulse form) and Q = 2 QijdqiÙdqj (a 2-form ). On the other hand, the functions of Pi and Ei are the components of a covector e = Eidqi - Pidv (set up by t) on TM; p = Q-PÙdT is a 2-d-form on R´M. The following particular special cases are distinguished: 1 a. If m = 2p and det(Qij) ¹ 0, then 2-form Q = 2 QijdqiÙdqj has the property h = Qp = det( Qij ) dq 1 ∧ ...dq m

is a d-form volume per M, set up by t. ⎛ Qij − Pi ⎞ ⎟ ≠ 0, det ⎜⎜ Pj 0 ⎟⎠ ⎝ b. If m = 2p + 1 and then 2-form p = Q-PÙdt has

269



















and the consequence (11.1.8 ') translates into the ''law'' associated with the system: EivI – PiFi = 0. All 2-forms, as (11.1.7), with condition verification (11.1.10) and satisfying Lorentz’s conditions (11.1.9), are called equivalent (admit the same characteristics). Their class is a dynamic notion. 1 Behavior of W form coefficients on a local map change on the basic manifold M. On a local map change on M and vector map respectively on TM, defined by formulas:



























π the property

p +1

=

⎛ Qij det ⎜⎜ ⎝ Pj

− Pi ⎞ 1 ⎟ dq ∧ ... ∧ dq m ∧ dt 0 ⎟⎠

that is a d-form volume on R´M.

11.3.10 Transforming superfield coefficients to a differentiable family of map changes on R × TM or a change of maps (set up by t) on the product manifold R´TM, given by formulas: ∂q i ∂q i q i = q i ( t , q h ), v i = h vh + , t = t, ∂t ∂q ∂q i ∂q i ∂q i ∂v i ∂v i dq i = h dq h + dt , dv i = h dv h + h dq h + dt . ∂t ∂t ∂ q ∂ q ∂ q (11.3.13) The superfield coefficients are subject to change formulas: ⎛ ∂q j ∂q j ⎞ ⎛ ∂q i ⎞ ∂v i ⎜ ⎟ 0 ⎜ h 0⎟ k ∂t ⎟ ⎜ ∂q ∂q h ⎜ ∂q ⎟ ⎜ ∂v j ∂v j ∂v j ⎟ ⎜ ⎟ ∂v i ⎛ ⎞ ⎛ Bhk − Akh E h ⎞ ⎜ 0 B − A E ⎟ ⎜ ⎟ 0 ij ij i ⎟ ⎜ ∂q k ∂v k ⎜ ⎟ ⎜ ∂t ⎟ ∂v h ⎟⎜ Qhk − Ph ⎟ ⎜ ∂q i ∂v i Qij − Pi ⎟ ⎜ 0 ⎜ Akh ⎟ ⎜ A ji 0 1 ⎟ 1⎟ ⎜ ⎜⎜ ⎟ ⎜− E ⎟ ⎜ ⎟ ⎟ ∂t Pk 0 ⎠ ⎝ ∂t Pj 0 ⎠⎝ k ⎝ ⎠ ⎝− E j ⎠. = It is apparent from these relationships that the functions Aij, Bij, and Qij change, as well as their counterparts in the formulas (11.3.12), and the functions Ei, and Pi are changed by: ⎞ ∂q i ⎛ ∂q j ∂v j Ph = h ⎜⎜ Pi − Aij − Qij ⎟⎟ , ∂t ∂t ∂q ⎝ ⎠ i i ∂q ∂q i ∂q j ∂v E h = h E i − h Pi + h Bij + ∂q ∂q ∂q ∂t

⎛ ∂v i ∂q j ∂q j ∂v i ⎞ ∂v i ∂v j ⎟ Aij + + ⎜⎜ h − h Qij ∂q ∂t ∂q ∂t ⎟⎠ ∂q h ∂t ⎝ (11.3.14) § 11.4 DYNAMIC SYSTEMS WRITTEN IN THE MAIN FORM (quasilinear) 11.4.1 The main form of equations of a dynamic system is TM.

In this subparagraph we will reconsider the general theory, in paragraph 0.4, where the basic manifold

It is known that the necessary and sufficient condition for a Newtonian dynamic system to be classical Lagrangean, is that Lagrange's form is closed and reduced. In this case Maxwell's equations are reduced to Helmholtz's equations. Assuming that a system of ordinary differential equations of order two (1.2.1) satisfies the condition ∂Fi = a ij2 = Aij ( t , q , q ) ∂q j , then it is of the form (1.11.1):

A ( t , q ,v )

B ( t , q ,v )

v + i (11.4.1) Fi(t,q, v , v ) = ij = 0, det(Aij)¹0 and it is called system of ordinary differential equations of order two, written in the main form. A system of equations (11.4.1) proper: det (Aij) ¹ 0, can be brought to kinematic form (Newtonian) j

j j (1.11.9): v = F ( t , q , v ) ,

where F

i

=-AihBH, where (Aij) is the inverse matrix of the matrix (Aij).

Being given the dynamic system (non-Lagrangean in classical sense) written in the main form

(11.4.1). From the definition of functions Fi and its properties, it follows: 11.4.2 Geometric interpretations

270

Aij =

∂Fi ∂v j , det (Aij) ¹ 0.





























On a local map change, on the M configuration manifold, given by (11.4.3), the coefficients in (11.4.1) are changed by formulas:

Ahk =

∂q i ∂q j ∂q i A , B = ij h ∂q h ∂q k ∂q h

11.4.2 Therefore, the functions Aij nondegenerate.

⎡ ⎤ ∂v j ∂q k B + Aij v p ⎥ , ⎢ i k p ∂q ∂q ⎣ ⎦

are the components of a two-fold covariant tensor, distinguished and

A ( v j − F i ) = 0

If the system (11.4.1) is written in the equivalent form ij = -AijF j and reciprocally we have Fi = - AijBj. The system (11.4.1) can also be written in the form Pfaff: 11.4.3

we deduct the coefficients Bi

Aij dv j + Bi dt = 0 , dq i − v i dt = 0 ,

11.4.3 Full form of Lagrange We will seek to associate the system (11.4.1) given a Lagrange form of the most general form (11.1.7), but which has as characteristic solutions of (11.4.1). This is the form:

Ω = ( A ji dv i + B j dt ) ∧ ( dq j − v j dt ) + 1 1 Bij ( dq i − v i dt ) ∧ ( dq j − v j dt ) + Q hk ( Ahi dv i + Bh dt ) ∧ ( Akj dv j + Bk dt ), 2 2

with ij

Bij + B ji = 0 , Q + Q

ji

= 0,

where Bij and Qhk anti-symmetrical coefficients are, for the time being,

undetermined. An immediate calculation leads to: W = A ji dv iÙdq j + [-(Bi + Bijvj)dqj+( - Ajivj + Q jhBh ) dvi]Ùdt + 1 1 (11.4.4) + 2 BijdqiÙdqj + 2 QhkAhiAkjdviÙdvj. formula that coincides with (11.1.7) if Lorentz's conditions are met: Noting: (11.4.5) Qij = QhkAhiAkj), We have the expressions (11.1.7), with: Ei = -(Bi + Bijvj), Pi = Aji(v j – Q jhBh). Now taking into account Bi = -AijF j, we have Lorentz's relations (11.1.2). The relationship (11.4.5), Q Q Q with the matriceal notation = (Qij) and Q = (Qij) is retranscribed in the form: = AT A. 11.4.4 The form of Lagrange reduced Suppose now that in the analytical expression of Lagrange's formW,, the coefficients Qij are null from Qij = AihAjkQhk, it follows that Qij = 0 and reciprocally. Based on relations (1.1.30), this property is independent of the local map chosen, it is therefore global. We therefore have the coefficients Aij, Bij, Ei =(Bi +Bijvj), Pi = Aijvj and Qij = 0. Such a form is called reduced Lagrange form. Let's consider the property of W form to be closed. From this property it follows that Maxwell's equations (1.1.30) are reduced to:

271





























∂Ahj ∂v i

−

∂Ahi ∂v j

= 0,

∂A jh ∂q i

−

∂Aih ∂q j

+

∂Bij ∂v h

= 0,

⎛∂ ∂B ∂ ⎞ ⎜ + vh ⎟ A − i + Bij = 0 , h ⎟ ij ⎜ ∂t ∂q ⎠ ∂v j ⎝ ∂B j ⎛∂ ∂B ∂ ⎞ ⎜ + vh ⎟B = i − , h ⎟ ij j ⎜ ∂t ∂q ⎠ ∂q ∂q i ⎝ ∂Bij ⎛ ∂Ahj ∂Ahi ⎞ h ⎜ ⎟ ⎜ ∂v i − ∂v j ⎟v + Aij − A ji = 0 , ∑ ∂q k = 0. ( i , j ,k ) ⎠ 11.4.6 ⎝ Let's remove the Bij functions from these equations. From the first group of equations and from the fifth it follows: I aij = aji, ∂Aih ∂A jh = j ∂v i Ii ∂v The third group, together with the second, leads to: ⎛∂ ∂B ∂ ⎞ Bij + ⎜⎜ + v h h ⎟⎟ Aij = ij ∂q ⎠ ∂v ⎝ ∂t , By changing indices i and j between them, it follows: ∂B j ⎛∂ ∂ ⎞ B ji + ⎜⎜ + v h h ⎟⎟ Aij = i . ∂q ⎠ ∂v ⎝ ∂t These two relationships, added and subtracted lead to:

∂Bi j

+

∂v III and respectively:

∂B j

⎛∂ ∂ ⎞ = 2⎜⎜ + v h h ⎟⎟ Aij ∂v ∂q ⎠ ⎝ ∂t i

1 ⎛ ∂Bi ∂B j ⎜ − 2 ⎜⎝ ∂v i ∂v i

⎞ ⎟. ⎟ ⎠ The fourth group of relationships, together with the sixth, leads to the relationship: ∂B j ⎛∂ ∂B ∂ ⎞ ⎜ + vh ⎟ Bij = i − ⎜ ∂t ∂q h ⎟⎠ ∂q j ∂q i ⎝ . The last two relationships say: ∂B j ⎞ ∂Bi ∂B j 1 ⎛ ∂ ∂ ⎞⎛ ∂B − i = ⎜⎜ + v h h ⎟⎟⎜⎜ ij − i ⎟⎟ j 2 ⎝ ∂t ∂q ∂q q ⎠⎝ ∂v ∂v ⎠ IV Thus, Maxwell’s equations I, II, III and IV are exactly Helmholtz's equations which comply with the assertion at the beginning of this subparagraph. It follows that the system (11.4.1) is self-adjoint and therefore Lagrangean. We have: Theorem 11.4.1 A system of equations of order two, written in the main form (11.4.1), is Lagrangean if and only if Lagrange form, written in a coreference adapted to the system is of reduced and closed canonical form . The general Lagrange form (written in the natural cobase), fulfilling Lorentz’s conditions, and closed, gives to the evolution space J1M = R´TM a presymplectic structure, if: Bij =

⎛ Bij det ⎜⎜ ⎝ A ji

− Aij ⎞ ⎟≠0 Qij ⎟⎠ ,

corresponding to the system (11.4.1). The coefficients (Aij, Bij, Qij, Ei, Pi) constitute the field associated with the system, with the significance (11.4.10), and the equations (4.1.10), field equations (Maxwell).

272









































From formulas (11.4.6) we have the functions Aij are the components of a proper 2-tensor and thus gives the configuration space a nonsymmetric, generalized Lagrange structure. § 11. 5 SUPERFIELDS ASSOCIATED WITH LAGRANGEAN DYNAMIC SYSTEMS In this paragraph we will consider the special case of classical Lagrangean dynamic systems. By customizing the general theory of superfields, we highlight fields associated with classical Lagrangean systems. A unitary theory is obtained. 11. 5.1 Superfields associated with general Newtonian dynamic systems 1 Lagrange form. Lorentz's conditions Being given a dynamic Newtonian system (modeled through): (11.1.0). It is associated with the equivalent system of equations: (11.1.2) and, with it, the 2-form i i i i i i of Lagrange-Gallissot: Ω G = = dv ∧ dq + ( F dq − v dv ) ∧ dt . The form Ω G accepts as characteristic trajectories of the dynamic system given. In general, a 2-form, (11.1.7), assuming that: (11.1.10), has the characteristics given by the formulas (11.1.8). As a consequence of these relationships (11.1.8), the relationship (11.1.8 ') results. The 11.1.2 proposition informs that the 2-form characteristics (11.1.7) coincide with the trajectories of the system (11.1.1) if and only if Lorentz's conditions are met: (11.1.9)), and the consequence (11.1.8 ') translates into the ''law'' associated with the system: Eivi – PiFi = 0. All 2-forms, as (11.1.7), fulfilling condition (11.1.10) and Lorentz’s conditions (11.1.9), are called equivalent (accept the same characteristic). Their class is a dynamic notion. For the behavior of field coefficients, Maxwell's principle, canonical isomorphism, volume element and differential operators, see section 11.11.1. 2 Fields associated with classical Lagrangean dynamic systems Proposition 11.5.2 A prerequisite for a dynamic order 2 system to be Lagrangean is that it is written in the main form: i  (11.5.1) F = A (t, q,v) v +B (t, q,v)=0, (v = q ). k

ki

k

It is proper if we have the property, considered by definition:

det(

∂Fi ) = det( Aij ) ≠ 0 , ∂v j

Such a system of equations, which shapes the dynamics of a real system, is not unique. We will say that two systems Fi = 0 and Gi = 0 are equivalent if they admit the same solutions. We will consider the j

j

j

j

class of systems Gi = FjD i , where D i =D i (t,q,v), with det(D i ) ¹ 0, "(t,q,v)ÎR´TM. Without restricting the generality, we can assume that, from the above-mentioned equivalence class, we have chosen a representative for which Aij = Aji. The functions Aij are the components of a d-metrics per M, as at a local map change, they are changed by formulas:

Ahk =

∂q i ∂q j Aij ∂q h ∂q k .

If (Aij) is the inverse matrix of (Aij), (11.1) implies, by multiplying with Ahi , the equations (11.5.1), for which: Fi = -AijBj. The coefficients –Bi = AijF j have the significance of covariant components of the field of forces Fi, in the metric Aij. We have: Proposition 11.5.3 The necessary and sufficient condition for a dynamic system, written in the form (2.1), to be Lagrangean is to be self-adjoint [2]. The conditions of the self-adjunction are:

det( Aij ) ≠ 0 ,

Aij =Aji

,

∂Aik ∂A jk = , ∂v j ∂v i

⎛∂ ∂Bi ∂B j ∂ ⎞ + i = 2⎜⎜ + v k k ⎟⎟ Aij , j ∂v ∂v ∂q ⎠ ⎝ ∂t 273

⎞ ⎟⎟ ⎠,

Proposition 11.5.4 A necessary and sufficient condition for a dynamic form system (11.5.6), to be equivalent to a self-adjoint system (Lagrangean), is that the 2-form of Lagrange (11.1.7) has the coefficients Qij º 0 [1]. Indeed, in this case Maxwell's equations are reduced to Helmholtz's equations. Let’s associate to the system (11.5.6) Pfaff forms:

θ i = dq i − v i dt , ψ = A dv j + B dt ,

ij i (11.5.1 ') i The corresponding 2-form of Lagrange is:

Ω =ψ i ∧θ i +

1 Bijθ i ∧ θ j 2 ,

(11.5.2) where Bij coefficients are, for the time being , unspecified. Unfolded, this 2-form is:

[

]

Ω = Aij dv i ∧ dq j + E i dq i − Pi dv i ∧ dt +

(11.5.2 ') Lorentz's conditions (11.5.4 ' ') are reduced to: E i + Bij v j = − Bi ,

1 Bij dq i ∧ dq j . 2

P − Aij v j = 0. 11.5.3 i Obviously, this 2-form accepts as characteristics the trajectories of system (11.5.6). Bij coefficients are obtained using Maxwell's equations. The first of the formulas (11.5.8) inform that the same force field, through its components– Bi, appears as a Lorentz field, relative to the family of field coefficients (Ei,Bij), the field of force being Fi = Aih(Eh+Bhkvk). The second informs that the functions Pi are covariant components of the speed field, relative to the metric Aij. We'll call them impulse functions. The F force field appears either of a mechanical (Newtonian) nature, contravariant (spray), or as a Lorentz force field, of electromagnetic nature, covariant. It can be considered as consequence of the field (E, B).

We retain the properties

det( Aij ) ≠ 0 ,

and Aij =Aji. Maxwell's equations are reduced to: ∂Bij ∂Ahi ∂Ahj ∂Aij ∂E j ∂P ∂Aih ∂A jh + − i + i + ij = − = h j j i ∂q ∂q = 0, ∂t ∂v ∂q ∂v (11.5.4) ∂v 0, ∂v 0, ∂Bij ∂E j ∂E i ∂Bij ∂Pj ∂Pi + − = = ∑ − j = i j h i ∂t ∂q ∂q ∂v 0, . ( i , j ,h ) ∂q 0, ∂v 0. Taking into account Lorentz's 2nd relationship (11.5.8), the last Maxwell equation, along with the property Aij =Aji , leads to the first relationship in (11.5.9). ∂B rotE + = 0, ∂t Relationships: 4th and 5th are written: divB = 0; they are the form of the wellknown Maxwell equations for the electric field Ei and magnetic induction Bij. The 2nd relationship connects the magnetic induction to the metric, and the 3rd, with Lorentz's 2nd condition, connects the electric field to the the metric. We will say that the subset of functions (Ei, Bij) constitute a field. Formulas for changing field coefficients are reduced to: ∂q i ∂q i ∂v i Ph = h Pi , E h = h E i − h Pi , ∂q ∂q ∂q

Ahk = 11.5.5

∂q i ∂q j Aij , ∂q h ∂q k 274









































∂Bi ∂B j 1 ⎛ ∂ ∂ ⎞⎛ ∂B ∂B j − i = ⎜⎜ + v k k ⎟⎟⎜⎜ ij − i j 2 ⎝ ∂t ∂q ∂q ∂q ⎠⎝ ∂v ∂v































Bhk =

∂q i ∂q j ∂v i ∂q j ∂v i ∂q j B + ( − ) Aij . ij ∂q h ∂q k ∂q k ∂q h ∂q h ∂q k

Observations The superfield, in this particular case, consists of two components: 1 A geometric component comprising a d-covector, impulse Pi and a d-tensor of second order, twice covariant, symmetrical and nondegenerate: Aij, which equips the configuration space with a Lagrange structure. 2 A "field" component (Ei, Bij), which fulfills the classical equations of the electromagnetic field. 3 Field (Ei,Bij), together with the metric of Aij, leads to the formulation of a field theory (classical), fulfilling Maxwell's equations 4 Taken together, coefficients Ei and Pi can be interpreted as components of a covector field on TM: Eidqi-Pidvi, this form Pfaff, based on the law of evolution, leads to the law (Ei - AijFj)vi = 0. 5 The field of force Fi, regarded as a semispray, allows the association to structuring the space, of a nonlinear connection. 6 Changing the W form coefficients , on a change of local coordinates on M (respectively on TM), through formulas (11.5. x), suggests the interpretation of the superfield as realizing a unified electrogravitational theory. 7 This view can be completed by considering the change of coordinates on the R´TM, as (11.5.5 '), in which case the formulas are obtained: ⎞ ∂q i ⎛ ∂q j Ph = h ⎜⎜ Pi − Aij ⎟⎟ , ∂t ∂q ⎝ ⎠ i i ⎛ ∂v i ∂q j ∂q j ∂v i ⎞ ∂q ∂q i ∂q j ∂v ⎟ Aij E h = h E i − h Pi + h Bij + ⎜⎜ h − h ⎟ ∂ t ∂ t ∂q ∂q ∂q ∂t ∂ q ∂ q ⎝ ⎠ (11.5.5 ') 8 From the relationships in the observation in subsection 4.1.5, in the condition Qij = 0, we deduct for the field the inverse matrix components: (Aij) is the inverse of the matrix (Aij), and the Bij functions are the contravariant components of Bij, raised with the components of the inverse matrix (Aij). 3 If the dynamic system is given by Lagrange function Being given the function of Lagrange L = L(t,q,v). Euler-Lagrange's equations: d ⎛ ∂L ⎞ ∂L =0 ⎜ ⎟− dt ⎝ ∂v i ⎠ ∂q i , (dqi = vidt), unfolded, become:

∂2L ∂2L ∂2L ∂L j j  v + v + − i = 0. i j i j i ∂v ∂v ∂v ∂q ∂v ∂t ∂q

They are of the main form (11.5.6), in which: ∂2L Aij = i j , ∂v ∂v ∂2L ∂2L ∂L Bi = i j v j + i − i . ∂v ∂q ∂v ∂t ∂q Lagrange's form is written:

1 ⎛ ∂2L ∂2L ∂2L i j ⎜ + − Ω L = i j dv ∧ dq + 2 ⎜⎝ ∂q i ∂v j ∂v i ∂q j ∂v ∂v

⎞ i ⎟⎟dq ∧ dq j . ⎠

⎡⎛ ∂L ⎤ ∂2L ∂2L ⎞ ∂2L + ⎢⎜⎜ i − v h h i − i ⎟⎟dq i − v h h i dv i ⎥ ∧ dt ∂v ∂q ∂v ∂t ⎠ ∂v ∂v ⎣⎝ ∂q ⎦ .

Thus, the following expressions are derived from the field components:

Aij =

∂2L , (metric), ∂v i ∂v j

Pi =

∂2L j ∂L ∂ 2 L ∂2L h v ( impulse ), E = − − v , ( electric. field ), i ∂v i ∂v j ∂q i ∂v i ∂t ∂v h ∂q i

Bij =

∂2L ∂2L − , ( magnetic ⋅ induction), ∂q i ∂v j ∂v i ∂q j

275

























values that obviously fulfill Maxwell's equations. The field of forces, in its Lorentz expression: Fi = Aih(Eh+Bhkvk), and based on Lagrange's equations,

∂L d ⎛ ∂L − ⎜ ∂q i dt ⎜⎝ ∂q i

⎞ ⎟⎟ + F h Ahi = Aih F h ⎠ , which confirms that Lorentz force Fi is the covariant

leads to: Fi = expression of the Newtonian field F

i

, based on the canonical d-metric Aij .

§ 11.6 GALLISSOT-SOURIAU FORMALISM IN PREFERENTIAL REFERENCES Once the dynamic Newtonian system (11.1.0) given, it implies the existence and use of a preferential coreference to the system, defined by (11.6.5), thanks to which some results are given a simplified form. We note a local map on M with coordinates (qi), and ( U ,Φ ) respectively the corresponding

(U , ϕ )

vector map on TM with coordinates (qi, vi). 11.6.1 DYNAMIC DIFFERENTIAL SYSTEMS WRITTEN IN KINEMATIC FORM 1 Preferential coreferences for dynamic differential systems written in kinematic form Forms Pfaff q i = dqi – vidt are called contact forms. Definition 11.6.1 We call non-autonomous semispray a field of vectors S defined on the product manifold R× TM that fulfills the characteristic properties: i (11.6.1) dt (S ) = 1, θ (S ) = 0

.

(

Given (U , ϕ ) a local atlas map , possibly complete, on M with coordinates (qi), and U ,Φ

)

respectively corresponding vector map on TM with coordinates (qi, vi). The vector atlas, with the local

coordinates (qi, vi), is called natural (it is not complete). The expression of the semispray S is: ∂ ∂ ∂ S = + v i i + F i i , F i ∈ C ∞ ( R × TM ). ∂t ∂q ∂v 11.6.2

The differential equations of the S -field trajectories on R× TM are:

dq i dv i dt = vi , = F i ( t , q , v ), = 1, dt dt 11.6.3 dt

and the second order differential equations of the above trajectories’ projections on M are: d 2qi dq = F i ( t ,q, ) 2 dt , (11.6.4) dt equations that shape the dynamics of a non-autonomous system. System (11.6.3) corresponds to Pfaff forms: i i i i i i (11.6.5) θ = dq − v dt , ψ = dv − F dt ,

i i whose annulment defines it, for θ (S ) = 0 , ψ (S ) = 0 . Thanks to the existence of the semispray S, it is necessary on R×TM coreference ( dt , θ i ,ψ i ) , which we call adapted coreference of the semispray. The

change of coreferences (dt,dqi,dvi) ® (dt,qi,yi) is given by (11.6.5) and dt = dt. i i 1 m Given now a change of local maps on manifold M, defined by the formulas: q = q ( q ,..., q ) . This induces a change of corresponding vector maps on TM by

q i = q i ( q h ), v i = formulas

∂q i j v . ∂q j By reference to vector maps, the S -field components are transformed by

vi = (11.6.6)

∂q i j ∂v i j ∂v i j i v , F = v + jF ∂q j ∂q j ∂v . 276





























i

i

dq i =

∂q i h ∂v i ∂v i h i h dq , d v = dq + dv ∂q h ∂q h ∂v h .

The differentials of functions q and v are: Calculating the forms q i, y i in the two maps and using (11.6.6) we obtain:

θi = (11.6.7) or matrix

∂q i h ∂v i h ∂v i h i θ , ψ = θ + hψ . ∂q h ∂q h ∂v ⎛ ∂q h ⎜ ⎛ θ h ⎞ ⎜ ∂q i ⎜ h⎟= ⎜ψ ⎟ ⎜ ∂v h ⎝ ⎠ ⎜ i ⎝ ∂q

⎞ 0 ⎟ i ⎟⎛⎜ θ ⎞⎟ h ⎟⎜ ∂v ⎝ψ i ⎟⎠ ⎟ ∂v i ⎠ .

2 Complete canonical form of Lagrange form, associated with a dynamic system A 2– Lagrange form (11.1.1) can be formally deduced with the help of a pseudotensor (semi d-tensor)

B − Aij ⎞ 1 ⎛⎜ ij ⎟ ⎜A ⎟ Q ji ij ⎝ ⎠ , by formula: covariant twice antisymmetrical T, of components T = 2 ⎛ Bij − Aij ⎞⎛ θ j ⎞ 1 1 1 ⎟⎜ j ⎟ = A jiψ i ∧ θ j + Bijθ i ∧ θ j + Qijψ i ∧ ψ j , Ω = ( θ i ,ψ i ) ∧ ⎜⎜ ⎜ ⎟ ⎟ 2 2 2 ⎝ A ji Qij ⎠⎝ψ ⎠

(11.6.8) which does not contain the differential dt. We will name the expression (11.6.8) the canonical form of the 2-form of Lagrange Ω. With the help of formulas (11.6.5) the expression (11.1.1) is also obtained:

Ω = A ji dv i ∧ dq j + [( − Bij v j + Aij F j )dq i − 1 1 − ( A ji v j + Qij F j )dv i ] ∧ dt + Bij dq i ∧ dq j + Qij dv i ∧ dv j , 2 2 (11.6.8')

which highlights Lorentz's conditions (11.1.3). Proposition 11.6.2 Being given a 2-form (defined on R×TM, closed or not and of 2m rank) as in (11.1.1), with the properties det(Aij)¹0 and (11.1.2), there is a unique spray S so that Ω is a Lagrange form associated with S. Demonstration From relationships (11.1.3) we will express the S field in the explicit form: (11.6.9)

F i = A ij ( E j + B jh v h ).

Including in (11.1.2) functions E i and Pi given through (11.1.3) we obtain for Ω expression (11.6.8), and the equations of its characteristics can be written in the form:

Bijθ j − Aijψ j = 0 , A θ j + Q ψ j = 0,

ij (11.6.10) ji which, assuming that (11.1.4) is fulfilled, leads to the equations (11.6.3). Theorem 11.6.3 A necessary and sufficient condition for a 2-form Ω (given by (11.1.1)) to be a Lagrange form associated with a spray S (given) is that its expression in a coreference adapted to S to be reduced to the canonical form (11.6.7). Demonstration Being given a dynamic system (11.6.3), it corresponds to a preferential coreference of the system, on R´TM: (dt,q i,y i), defined by: (11.6.5), of which, through the inverse, we have: i i i i i i (11.6.5') dq = θ + v dt , dv = ψ + F dt . We get to the expression:

1 1 Ω = A jiψ i ∧ θ j + Bij θ i ∧ θ j + Qijψ i ∧ψ j + ( E i − Aij F j + Bij v j )θ i ∧ dt 2 2 − ( Pi − A ji v j − Qij F j )ψ i ∧ dt ,

(11.6.11) Being now a 2-form Lagrange W of general form. Taking into account (11.6.9), the form W gets the expression (11.6.11).

277





























11.6.2 PREFERENTIAL COREFERENCES FOR DYNAMIC SYSTEMS DIFFERENTIAL WRITTEN IN THE MAIN FORM

∂Fi = aij0 = A ji ( t , q , q ) j 1 The main form of the evolution equations If ∂v , then: j A ( t , q , v ) v + Bi ( t , q , v ) = 0, (11.6.12) F (t,q, v , v ) = ij i

is called system of ordinary differential equations of order two, written in the main form. A proper system of equations: det (Aij) ¹ 0, can be brought to the kinematic form (Newtonian): i v ʹ j = F ( t , q , v ) , where Fi = -BhAhi.

The expression of the semispray S , defined by relationships (11.6.1): dt(S)=1, qi(S) = 0,yi(S) S=

∂ ∂ ∂ + v i i − Bh A hi i ∂t ∂q ∂v , where (Aij) is the inverse matrix of (Aij). We have at the same

= 0, is time the cobase (dt,qi,yi), which is called cobase adapted to the system:

2 Lagrange form in general, (in adapted cobase). Its construction The system (11.6.6) can be written in equivalent form (as Pfaff system on TM): θ i = dq i − v i dt = 0 ,

ψ = A ji ( t , q , v )dv i + B j ( t .q .v )dt = 0. (11.6.13) i We will seek to associate it with a Lagrange form, the most general form, but which has as characteristics, solutions of (11.6.13). This is : W = (Ajidv i+Bjdt)Ù(dq j-v jdt) + 1 1 i i j j 2 Bij(dq -v dt)Ù(dq -v dt) + 2 Qhk(Ahidvi+Bhdt)Ù(Akjdvj+Bkdt),

where the anti-symmetric Bij and Qhk coefficients are, for the time being, unspecified. An immediate calculation leads us to: A ji dv i ∧ dq j − ( Bi + Bij v j )dq i ∧ dt ( A ji v j − Q hk Ahi Bk )dv i ∧ dt W= + 1 1 2 Bijdqi Ùdqj + 2 QijdviÙdvj,

QhkAhiAkj,

where Qij = Pi = A ji v j − Q hk Ahi Bk

a formula that coincides with (11.1.1), if

Ei = − Bi − Bij v j

,

and Bi =-aihFH. In the adapted cobase, W form is written canonically: − δ ij ⎞⎛ θ j ⎞ 1 ⎟⎜ ⎟ Qij ⎟⎠⎜⎝ψ j ⎟⎠ =yiÙq i+ 2 Bijq iÙq j +QijyiÙyj.

⎛ Bij θ ,ψ i ∧ ⎜⎜ ⎝ δ ij W=

(

i

)

3 Geometric interpretation If a map change is carried out, then from (9.4) and from (10.1) it

follows that coefficients Aij and Bij change according to formulas:

∂q i ∂q j ∂q i ∂q i ∂q j ∂v k p Ahk = Aij Bh = h Bi − h Aij v ∂q ∂q ∂q k ∂q p . ∂q h ∂q k , A dynamic system given, through the Newtonian equations, can be described by any other C jF = 0 system of equivalent equations, as: i j . On a local map change, forms (11.5.2) change according to formulas:

θ

h

=

∂q h i ∂q i θ , ψ = ψi. h ∂q h ∂q i

The shift from the canonical form, written in a preferential reference, to the one written in another, is done with change of coefficients:

278







































∂q i ∂q j ∂q h ∂q k ij hk B Q = Q ij ∂q h ∂q k ∂q i ∂q j (11.6.14) , . Formulas (11.6.14) inform that: Theorem 11.6.7 In the system-adapted atlas (11.6.12), the functions Bij and Qij are respectively the components of a two-fold covariant tensor and those of a tensor twice contravariant Definition 11.6.8 We say that the form W is the reduced canonical form if, reported to the reference i (dt, q , yi),is: 1 (11.6.15) W = y i Ùq i + 2 Bijq iÙq j, Bij + Bji = 0. Its characteristics are given by: q i = 0,yi - Bijq j = 0, Bhk =

system that obviously has the solution q i = y i = 0. If we include in (11.6.3) the expressions of q reference, the expression:

i

and y

i

, given by: (11.5.2), we find in the natural

1 W = AijdqiÙdqj + [(Bi – Bijvj)dqi + Ajivjdvi]Ùdt + 2 BijdqiÙdqj. To this system we associate the Pfaff system: θ i = dq i − v i dt , ψ i = Aij dv j + Bi dt . (11.6.16) The expression of spray S, defined by relationships: dt(S)=1,q i(S)=0,(yi(S) = 0), is: ∂ ∂ ∂ S = + v i i − A ih Bh i ∂t ∂q ∂v , ij where matrix (A ) is the inverse of matrix (Aij). Associated with the system (11.5.1) and the adapted reference (dt, q i, yi).

4 Reduced canonical Lagrange form Definition 11.6.4 We will call reduced canonical form of a 2- W form the expression: 1 (11.6.17) W = yiÙq i + 2 Bijq iÙq j, with Bij +Bji = 0. Such form shall be formally obtained by:

⎛ Bij 1 Ω = ( θ i ,ψ i ) ∧ ⎜⎜ 2 ⎝ δ ij (11.6.18)

− δ ij ⎞⎛ θ j ⎞ ⎟⎜ ⎟. 0 ⎟⎠⎜⎝ψ j ⎟⎠

Proposition 11.6.5 The characteristics of 2-form W coincide with the trajectories of system (10.1). Indeed, we have: ∂Ω ∂Ω =θ i = −ψ i + Bijθ j , i ∂ψ i ∂θ , system that has the solution q i = yi = 0. Replacing in (11.6.13) the expressions of q obtained in the natural coreference (dt, dqi, dvi):

i

and y i, given by (11.6.5), the local Wexpression is

1 (11.6.19) W = AjidvJÙdqi – [(BijvJ+ Bi) dqi + Aij vJdvi]Ùdt + 2 BijdqiÙdqJ. It is of general form: (11.1.7), in which: Ei =-(Bi -Bijv j), Pi = Ajiv j, Qij=0. If Aij = d ij, then Pi = vi, Bi =-Fi, and form W has the expression given by Souriau W S. If Bij = 0, the expression of Gallissot is obtained. to:

Let's assume now that the 2-form W is closed. In this case Maxwell's equations (11.11.7) are reduced

279

∂Ahjh ∂Aih ∂Bij ∂Ahi = 0, − = h , j ∂v ∂v ∂q i ∂q j ∂v ∂Aij ∂B jh ∂B j ∂A − vh − B ji − i + v h hij = 0 , i ∂t ∂v ∂q ∂q i

−

∂Bij

∂B jh ⎞ h ∂Bi ∂B j ⎛ ∂B ⎟v + j − i = 0 , + ⎜⎜ ihj − ∂t ⎝ ∂q ∂q i ⎟⎠ ∂q ∂q ∂Ah j ∂Ah i ∂Bij − + Aij − A ji = 0 , ∑ = 0. i j k ∂ v ∂ v ( i , j ,k ) ∂q (11.6.20)

Let's remove the Bij functions from these equations. Of the first group of equations and the fifth we have: ∂Ahj ∂Ahi = . i ∂v j (11.6.21) Aij= Aji, ∂v The third group leads us to: ⎛∂ ∂B ∂ ⎞ ⎟A − i Bij = ⎜⎜ + v k k ⎟ ij ∂q ⎠ ∂v j ⎝ ∂t . By exchanging indices i and j, adding, respectively subtracting the two formulas, we have:

∂B j and

(11.6.22)

∂q

i

+

⎛∂ ∂Bi ∂ ⎞ = 2⎜⎜ + v k k ⎟⎟ Aij j ∂q ∂q ⎠ ⎝ ∂t 1 ⎛ ∂B j ∂Bi ⎜ − 2 ⎜⎝ ∂v i ∂v j

⎞ ⎟ ⎟ ⎠. The fourth group of relationships (10.6), using relationships (10.8), becomes: ⎛∂ ∂B j ∂Bi ∂ ⎞ ⎜ + vk ⎟B = − k ⎟ ij ⎜ ∂t ∂q ⎠ ∂q i ∂q j , ⎝ so that from the last two groups of equations we have: ∂B ∂B j ∂Bi 1 ⎛⎜ ∂ + v k ∂ ⎞⎟ ⎛⎜ j − ∂Bi ⎞⎟ − ⎜ i ∂q k ⎟⎠ ⎜⎝ ∂v i ∂v j ⎟⎠ ∂q j = 2 ⎝ ∂t (11.6.23) ∂q . Equations (11.6.21). (11.6.22) and (11.6.23) are Helmholtz's equations, which inform that the system (11.6.12) is Lagrangean. We have: Theorem 11.6.6 A system of equations of order two, written in the main form (11.6.12), is Lagrangian if and only if its Lagrange form written in the system-adapted reference can be of reduced and closed canonical form. Bij =

⎛ Bij ⎜ ⎜A The symmetrical part of the T-tensor: ⎝ ji i

− Aij ⎞ ⎟ Qij ⎟⎠

⎛ 0 ⎜ ⎜A is: ⎝ ( ij )

j

i

A( ij ) dx ⊗ dx − A( ij ) dv ⊗ dv

− A( ij ) ⎞ ⎟ 0 ⎟⎠

. It leads to the square form:

j

. which is reduced to the canonical form if and only if the conditions (11.1.9) are met. A similar approach to that in the preceding paragraph may also be considered in the case of dynamic systems in the main form. The shift from the canonical form, written in an adapted referential, to the one written in another, is done with the change of coefficients:

Qhk

∂q i ∂q j ∂q h ∂q k ij hk = h Q Q = Q ij ∂q ∂q k ∂q i ∂q j , .

(11.6.24) In the adapted cobase, W form is written canonically:

280





























∂Ahj





⎛B 1 i θ ,ψ i ∧ ⎜ iji ⎜δ 2 ⎝ j W=

(

)

− δ i j ⎞⎛ θ j ⎞ ⎟⎜ ⎟ Q ij ⎟⎠⎜⎝ψ j ⎟⎠

1 1 =q iÙyi+ 2 Bijq iÙq j + 2 QijyiÙyj ,

called the canonical form of the dynamic system written in the main form. Reciprocally, being given a set of functions Bij(t,q,v) and Qij(t,q,v), antisymmetric (Bij+Bji=0, Q ij+Q ji=0) so that:

⎛B det ⎜ iji ⎜δ ⎝ j

− δi j ⎞ ⎟≠0 Q ij ⎟⎠ ,

otherwise arbitrary. With them we build the form of Lagrange: 1 1 W = yiÙq i + 2 Bijq iÙq j + 2 QijyiÙyj , the characteristics of which are the trajectories of the system.

281

XII OPTIMAL DYNAMIC SYSTEMS

Control theory is a superior stage in the dynamic system theory and deals with the control of continuously operating dynamic systems. For such systems, the influence of awareness in their development is essential. The law of their evolution is much more complex, being closely tied to free will. This chapter proposes to present to the reader some basic and elementary notions of the theory of dynamic systems with control § 12.1 GENERAL OPTIMIZATION PROBLEMS FOR SOME DYNAMIC SYSTEMS One of the applications of differential calculation is the study of the behavior of a function in a convenient vicinity of an arbitrary point in its definition domain. As an application of this for real value functions, it has been possible to successfully address the optimization problems commonly encountered in practice. It is known that, for small variations of the argument, the variation of a function can be approximated by the difference of the latter (the linear part of the increase). The accuracy with which it approximates the variation, and in what situations it can be applied, will be studied. It is clear that such an approximation is meaningless at those points where the function derivative is canceled. Truthfully, given f:D⊂Rm→Rn a differentiable function in a point xo∈D, in which: (12.1.1) f '(xo) = 0. In this situation, whichever the variation of the argument dx∈R m, the variation of the function will be dy = 0. The points in the domain of definition of the function f that have this property bear the name critical points for said function f. To give some examples of critical points, in the case of scalar differentiable functions, f:Rm→Rn, the critical points are the local extrema points of f , meaning those with xo∈ D for which there is a vicinity U x o = U(xo) with the property: (12.1.2) f(xo) ≤ f(x), (or f(xo) ≥ f(x)), (∀x∈U(xo)). Truthfully, if U(xo) = B(xo,r) is the inside of a sphere with the centre xo and radius r, f being differentiable in xo, it means that: x − xo (12.1.3) f(x)-f(xo)=f'(xo)(x-xo)+ O(x-xo), ∀x∈B(xo,r). x−x

o O(x-x ), which is of order ≤ r2, and with the For a small enough r, neglecting in point (12.1.3) o condition (12.1.2), arises the property that local extrema points check one of the following variational inequalities: f'(xo)(x-xo) ≥ 0 or f'(xo)(x-xo) ≤ 0, ∀x∈B(xo,r). Considering that f is a function with a real variable

x−x

o < r, implies the (m=1), and xo is a local minimum (local maximum), the inequality f'(xo)(x-xo)≥ 0,∀ inequalities –f'(xo)h ≤ 0 and f'(xo)h ≥ 0, for any h∈(0,r), which can only occur if f'(xo)=0. Take f:D⊂Rm→Rn a differentiable function in xo∈D. For any e = (η1,...,ηm) ∈ Rm, having the df ( x0 ) property Σ(ηi)2=1, there is de , therefore: m ∂f df ( x0 ) = ∑ i ( x o )η i de i =1 ∂x . df d f ( x o1 + tη 1 ,..., x om + tη m ) Truthfully, by definition there is de (x0) = dt / t=o, wherein xo= (x1o,...,xmo). Considering xi=xio+tηi, i= 1, m , and applying the chain rule, i.e. the formula for computing the derivative of

m ∂f df ( x0 ) = ∑ i ( x o )η i i =1 ∂x a composition of two of more functions, the result is: de , because for t=0 there is xi=xio and (xi)'=ηi, (i= 1, m ). Be there a local extremum, it is now a local extremum on any straight line passing through it and, because f is differentiable in x0, then a necessary extrema condition asks that Férmat’s equations be satisfied:

∂f ∂x i

( x o ) = 0 , ( i = 1, m ),





282

























CH.

12.1.1 The extrema of a function In many problems, both in terms of theory as well as application, differentiable functions are considered f:D⊂Rm→R. Take A as a set with the property A ⊂ D. The problems of optimizing (minimizing) lead to determining the points xo∈A, such that: (12.1.2') f(xo) ≤ f(x), ∀ x∈A. If set A is compact in Rn, Weierstrass’ theorem states that there exists at least one solution xo to the problem (12.1.2'), as in: f(xo) = inf{f(x) | x∈A}. The following situations in the study stand out: U 1o. If x0∈IntA, there exists a vicinity xo ⊂ A and xo is among the solutions of Férmat’s equation (12.1.1) which, as mentioned above, is a necessary condition for the local extrema, the points xo, so that f'(xo) = 0, being critical. 2o If xo∈∂A (the border of set A), then xo is a solution to the inequation: (12.1.4) f''(xo)(x-x0)≥0, ∀x∈A, which can no longer be reduced to Férmat’s equation. To determine the sufficient conditions for the extrema, in case 1o, we will assume that the function f belongs to a differentiability class of at least two, in which case Taylor’s formula can be written: 1 Δf(x) = f(x) – f(xo) = df(x0) + 2 d2f(ξ), where ξ = xo + θ(x-xo), with 0