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Introduction to the

Geometry of Classical Dynamics

Renato Grassini

Dipartimento di Matematica e Applicazioni

Universita di Napoli Federico II

HIKARI LTD

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HIKARI LTD

Hikari Ltd is a publisher of international scientific journals and books.

www.m-hikari.com

Renato Grassini, Introduction to the Geometry of Classical Dynamics, Firstpublished 2009.

No part of this publication may be reproduced, stored in a retrieval system,

or transmitted, in any form or by any means, without the prior permission ofthe publisher Hikari Ltd.

ISBN 978-954-91999-4-9

Typeset using LATEX.

Mathematics Subject Classification: 70H45, 58F05, 34A09

Keywords: smooth manifolds, implicit differential equations, dAlembertsprinciple, Lagrangian and Hamiltonian dynamics

Published by Hikari Ltd

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iii

Preface

The aim of this paper is to lead (in most elementary terms) an under-graduate student of Mathematics or Physics from the historical Newtonian-dAlembertian dynamics up to the border with the modern (geometrical) Lagrangian-Hamiltonian dynamics, without making any use of the traditional (analytical)formulation of the latter. 1

Our expository method will in principle adopt a rigorously coordinate-freelanguage, apt to gain from the very historical formulation the conscious-

ness (at an early stage) of the geometric structures that are intrinsic to thevery nature of classical dynamics. The coordinate formalism will be confined tothe ancillary role of providing simple proofs for some geometric results (whichwould otherwise require more advanced geometry), as well as re-obtaining thelocal analytical formulation of the theory from the global geometrical one. 2

The main conceptual tool of our approach will be the simple and generalnotion of differential equation in implicit form, which, treating an equation

just as a subset extracted from the tangent bundle of some manifold through ageometric or algebraic property, will directly allow us to capture the structuralcore underlying the evolution law of classical dynamics. 3

1 Such an Introduction will cover the big gap existing in the current literature betweenthe (empirical) elementary presentation of Newtonian-dAlembertian dynamics and the (ab-stract) differential-geometric formulation of Lagrangian-Hamiltonian dynamics. Standardtextbooks on the latter are [1][2][3][4], and typical research articles are [5][6][7][8][9][10].

2 The differential-geometric techniques adopted in this paper will basically be limited tosmooth manifolds embedded in Euclidean affine spaces, and are listed in Appendix (whosereading is meant to preceed that of the main text). More advanced geometry can be foundin the textbooks already quoted, as well as in a number of excellent introductions, e.g.[11][12][13][14].

3 Research articles close to the spirit of this approach are, among others, [15][16] (onimplicit differential equations) and [17][18][19][20] (on their role in advanced dynamics).

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Contents

Preface iii

1 From Newton to dAlembert 1

1.1 The data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Configuration space . . . . . . . . . . . . . . . . . . . . . . . . . 1Mass distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Force field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Mechanical system . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 The question . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Smooth motions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Dynamically possible motions . . . . . . . . . . . . . . . . . . . . 31.3 The answer after Newton . . . . . . . . . . . . . . . . . . . . . . 4

Newtons law of constrained dynamics . . . . . . . . . . . . . . . . 41.4 dAlemberts reformulation . . . . . . . . . . . . . . . . . . . . . 4

dAlemberts principle of virtual works . . . . . . . . . . . . . . . . 41.5 dAlemberts implicit equation . . . . . . . . . . . . . . . . . . . 5

Tangent dynamically possible motions . . . . . . . . . . . . . . . . 5dAlembert equation . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 From dAlembert to Lagrange 8

2.1 Integrable part of dAlembert equation . . . . . . . . . . . . . . 8Restriction of dAlembert equation . . . . . . . . . . . . . . . . . . 9Extraction of the integrable part . . . . . . . . . . . . . . . . . . . 9

2.2 Lagrange equation . . . . . . . . . . . . . . . . . . . . . . . . . 10Covector formulation . . . . . . . . . . . . . . . . . . . . . . . . . 10Riemannian geodesic curvature field . . . . . . . . . . . . . . . . . 11Normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Integral curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Classical Lagrange equations . . . . . . . . . . . . . . . . . . . . . 17

iv

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Table of Contents v

2.3 Euler-Lagrange equation . . . . . . . . . . . . . . . . . . . . . . 23Conservative system . . . . . . . . . . . . . . . . . . . . . . . . . 23Lagrangian geodesic curvature field . . . . . . . . . . . . . . . . . . 24Integral curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Classical Euler-Lagrange equations . . . . . . . . . . . . . . . . . . 25

2.4 Variational principle of stationary action . . . . . . . . . . . . . 26Variational calculus . . . . . . . . . . . . . . . . . . . . . . . . . 26Variational principle . . . . . . . . . . . . . . . . . . . . . . . . . 29Riemannian case . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 From Lagrange to Hamilton 33

3.1 Legendre transformation . . . . . . . . . . . . . . . . . . . . . . 33Lagrangian function and Legendre transformation . . . . . . . . . . 33Coordinate formalism . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Hamiltonian function . . . . . . . . . . . . . . . . . . . . . . . . 35Energy and Hamiltonian function . . . . . . . . . . . . . . . . . . 35Coordinate formalism . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Hamiltonian vector field . . . . . . . . . . . . . . . . . . . . . . 36Canonical symplectic structure and Hamiltonian vector field . . . . . 36Coordinate formalism . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 Hamilton equation . . . . . . . . . . . . . . . . . . . . . . . . . 37

Cotangent dynamically possible motions . . . . . . . . . . . . . . . 37Hamilton equation . . . . . . . . . . . . . . . . . . . . . . . . . . 37Classical Hamilton equations . . . . . . . . . . . . . . . . . . . . . 38

4 Concluding remarks 39

4.1 Inertia and force . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . 404.3 Geometrizing physical fields . . . . . . . . . . . . . . . . . . . . 414.4 Geometrical dynamics . . . . . . . . . . . . . . . . . . . . . . . 41

5 Appendix 42

5.1 Submanifolds of a Euclidean space . . . . . . . . . . . . . . . . 42Affine subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Embedded submanifolds . . . . . . . . . . . . . . . . . . . . . . . . 43Atlas of charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Locally Euclidean topology . . . . . . . . . . . . . . . . . . . . . . 44Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Smooth curves and tangent vectors . . . . . . . . . . . . . . . . . . 44Tangent vector spaces . . . . . . . . . . . . . . . . . . . . . . . . 45Open submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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vi Table of Contents

Implicit function theorem . . . . . . . . . . . . . . . . . . . . . . 475.2 Tangent bundle and differential equations . . . . . . . . . . . . . 49Tangent bundle and canonical projection . . . . . . . . . . . . . . . 49Vector fibre bundle structure . . . . . . . . . . . . . . . . . . . . . 49Tangent lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Differential equations in implicit form . . . . . . . . . . . . . . . . 50Integral curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Integrable part . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Vector fields and normal form . . . . . . . . . . . . . . . . . . . . 51Cauchy problems and determinism . . . . . . . . . . . . . . . . . . 51

5.3 Second tangent bundle and second-order differential equations . 52

Second tangent lift and second tangent bundle . . . . . . . . . . . . 52Affine fibre bundle structure . . . . . . . . . . . . . . . . . . . . . 53Second-order differential equations in implicit form . . . . . . . . . 55Integral curves and base integral curves . . . . . . . . . . . . . . . 56Semi-sprays and normal form . . . . . . . . . . . . . . . . . . . . 57Cauchy problems and determinism . . . . . . . . . . . . . . . . . . 58

5.4 Cotangent bundle and differential forms . . . . . . . . . . . . . 58Cotangent vector spaces . . . . . . . . . . . . . . . . . . . . . . . 58Cotangent bundle and canonical projection . . . . . . . . . . . . . . 59Vector bundle structure . . . . . . . . . . . . . . . . . . . . . . . . 59Differential 1-forms . . . . . . . . . . . . . . . . . . . . . . . . . 59Semi-basic differential 1-forms . . . . . . . . . . . . . . . . . . . . 61Semi-Riemannian metric . . . . . . . . . . . . . . . . . . . . . . . 62Almost-symplectic structure . . . . . . . . . . . . . . . . . . . . . 63

5.5 Intrinsic approach to smooth manifods . . . . . . . . . . . . . . 64Intrinsic geometry of embedded submanifolds . . . . . . . . . . . . . 64Intrinsic geometry of smooth manifolds . . . . . . . . . . . . . . . 66

References 67

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Chapter 1

From Newton to dAlembert

In this chapter, we shall recall the problem of classical particle dynamics,Newtons answer to the problem and dAlemberts reformulation of the answer.The latter will then be shown to correspond to a differential equation in implicitform on a Euclidean space.

1.1 The data

Classical particle dynamics deals with an empirical problem, whose data in

the simplest cases can be described in mathematical terms as follows.

Configuration space

A reference space mathematical extension of a rigid body carrying an ob-server is conceived as a 3-dimensional, Euclidean, affine space E3 , modelledon a vector space E3 with inner product (time will be conceived as an ori-ented, 1-dimensional, Euclidean, affine space, classically identified with thereal line R ).

Particle is synonymous with point-like body, i.e. a body whose positionin

E3 is defined by a single point of

E3 .

Therefore, for a given (ordered) system of particles, a position or con-figuration in E3 is defined by a single point of E := E3 (3-dimensional,Euclidean, affine space, modelled on E := E3 ).

1

1 Recall that the inner product in E is defined by putting, for all v = (v1, . . . ,v) andw = (w1, . . . ,w) belonging to E,

v w :=

i=1

vi wi

1

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2 1 From Newton to dAlembert

The particle system may generally be subject in E3 to some holonomic (orpositional) constraints, owing to which it is virtually allowed to occupy onlythe positions belonging to an embedded submanifold

Q Ecalled configuration space of the system in E3 .

Q generally consists of all the points p E satisfying a number < 3of independent scalar equalities f(p) = 0 ( = 1, . . . , ), called two-sidedconstraints, and/or some strict scalar inequalities g(p) > 0 ( = 1, . . . , ),called strict one-sided constraints. Under usual hypotheses of regularity onthe constraints, Q is an embedded submanifold of

E, whose dimension n :=

dim Q = 3 called number of the degrees of freedom of the system isgiven by the dimension of the Euclidean environment minus the number ofthe two-sided constraints (in absence of two-sided constraints, Q is an opensubmanifold). 2

Mass distribution

The response of the system to any internal or external influence, will generallydepend on how massive its particles are, the inertial mass of a particle beingconceived as a positive scalar quantity.

The inertial mass distribution carried by the system will be denoted bym := (m1, . . . , m)

Force field

The force resultant of all the internal and/or external influencesacting in E3 on the particles and not depending on their being constrainedor unconstrained, is generally described as a smooth vector-valued mapping,called force field,

F : U

E

T

E E : (p, v)

F(p, v) = (F1(p, v), . . . , F(p, v))

where U denotes an open subset of E containing Q .Remark that, if F|{p}E = const. for all p U, then F called positional

force field can be regarded as a smooth mapping f : U E E : p f(p),where f(p) denotes the constant value of F|{p}E . 3

2 We shall not consider non-strict one-sided constraints, which would give rise to a mani-fold Q with boundary (nor shall we consider time-dependent constraints, which would giverise to a manifold Q fibred over the real line).

3 We shall not consider time-dependent force fields, which would later give rise to time-dependent differential equations.

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Mechanical system

The above empirical mechanical system will briefly be denoted by the triplet

S:= (Q,m, F)

1.2 The question

With reference to such a mechanical system

S, the basic problem of dynamics

can be expressed in the following terms.

Smooth motions

A smooth motion of the particle system in the reference space E3 is describedas a smooth curve of E, say

: I R E: t (t) = p(t)

Such a curve is meant to establish a configuration p(t) at each instant tof a time interval I R , along which the positions p(t) = (p1(t), . . . , p(t))of the particles, their velocities p(t) = (p1(t), . . . , p(t)) and their accelera-tions p(t) = (p1(t), . . . , p(t)) (as well as the derivatives of any order) varycontinuously.

Dynamically possible motions

Smooth dynamics basically deals with the time-evolution problem in theunknown expressed by the following question:

For the above constrained point-mass system, what are the smooth mo-tions in the chosen reference space that are possible under the action of thegiven ?

Such motions will briefly be called the dynamically possible motions (DPMs)of S (whereas the smooth motions which would be possible in absence of force,i.e. F = 0 , will be said to be the inertial motions of S).

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1.3 The answer after NewtonAfter Newton, the answer to the above predictive question is given by thefollowing law.

Newtons law of constrained dynamics

A smooth curve : I E: t (t) = p(t)

is a DPM of S, iff, for all t I, 4

p(t) Qm p(t) = F(p(t), p(t)) + (t)

(t) Tp(t)Q

The first condition just exhibits the kinematical effects of the constraints,which only allow motions living in Q .

The second condition is the classical Newtons law with a right hand sideencompassing the possible dynamical effects of the contraints, expressed byan unknown constraint reaction (t)

E.

The third condition expresses the only known empirical requisite of theconstraint reaction, which apart from possible frictions tangent to Q isalways orthogonal to Q. 5

1.4 dAlemberts reformulation

After dAlembert, the unknown constraint reaction can be cancelled fromNewtons law of constrained dynamics as follows.

dAlemberts principle of virtual works

The last two of the above conditions can obviously be expressed in the form

F(p(t), p(t)) m p(t) Tp(t)Q4 For any = (1, . . . , ) R and w = (w1, . . . ,w) E, we shall put w :=

(1w1, . . . , wn) . Moreover, for any p Q , Tp Q will denote the orthogonal complementin E of the tangent vector space TpQ .

5 If the empirical law of friction is to be taken into consideration, then you will embodyit in F .

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that is,

F(p(t), p(t)) m p(t)

p = 0 , p Tp(t)Qcalled dAlemberts principle of virtual works (since the inner product thereindefines the work ofactive force F(p(t), p(t)) and inertial force mp(t) alongany virtual displacement p , i.e. any infinitesimal displacement tangent toQ and therefore virtually allowed by the constraints). 6

So a smooth curve

: I E: t (t) = p(t)

is a DPM of S, iff it satisfies, for all t I the time-evolution lawp(t) Q , m p(t) p = F(p(t), p(t)) p , p Tp(t)Q ()

1.5 dAlemberts implicit equation

From the mathematical point of view, condition () shows that determiningthe DPMs of S is a second-order differential problem, whose unknown is asmooth curve of E. It turns into a first-order differential problem, whoseunknown is a smooth curve of TE, as follows.

Tangent dynamically possible motions

If : I E: t (t) = p(t)

is DPM of S, its tangent lift

: I TE: t (t) = (p(t), p(t))

will be called a tangent dynamically possible motion (TDPM) of S.6 Owing to footnotes 1 and 4, dAlemberts principle reads

i=1

Fi(p(t), p(t)) mipi(t)

pi = 0 , p = (p1, . . . , p) Tp(t)Q

Remark that, if Q is an open submanifold of E (absence of two-sided constraints), onehas Tp Q = E

= {0} for all p Q (absence of constraint reaction) and then dAlembertsprinciple simply reads

mipi(t) = Fi(p(t), p(t)) , i = 1, . . . ,

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DPMs and TDPMs bijectively correspond to one another, since the tangentlift operator is obviously inverted by the base projection operator . Trough such a bijection, the problem of determining the DPMsproves to be naturally equivalent to that of determining the TDPMs.

Owing to (), a smooth curve

c : I TE: t c(t) = (p(t), v(t))

is a TDPM of S, iff it satisfies, for all t I, the time-evolution law

p(t) Q , p(t) = v(t) , m v(t) p = F(p(t), v(t)) p , p Tp(t)Q ()

dAlembert equation

Condition () will now be seen to correspond to a first-order differential equa-tion in implicit form on TE (second-order on E), 7 namely dAlembert equa-tion

DdAl := {(p, v; u, w) T TE | p Q , u = v , F(p, v) m w Tp Q }= {(p, v; u, w) T TE | p Q , u = v , m w p = F(p, v) p , p TpQ } T2E

Proposition 1 The TDPMs of S are the integral curves of DdAl (and thenthe DPMs are its base integral curves).

Proof Recall that a smooth curve

c : I TE: t c(t) = (p(t), v(t))

is an integral curve of DdAl , iff its tangent lift

c : I T TE: t c(t) = (p(t), v(t); p(t), v(t))

satisfies conditionIm c DdAl

that is, for all t I,

c(t) = (p(t), v(t); p(t), v(t)) DdAlwhich is exactly condition (), characterizing the TDPMs.

7 Recall that TE= E E is a Euclidean affine space modelled on E E. Its tangentbundle is therefore T TE= (E E) (E E) .

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Also recall that a smooth curve

: I TE: t (t) = p(t)

is a base integral curve of DdAl , iff its second tangent lift

: I T TE: t (t) = (p(t), p(t); p(t), p(t))

satisfies conditionIm DdAl

that is, for all t I,

(t) = (p(t), p(t); p(t), p(t)) DdAlwhich is exactly condition (), characterizing the DPMs.

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Chapter 2

From dAlembert to Lagrange

Dynamics is now a problem of integration, i.e. determination and/or qualita-tive analysis of the integral curves of dAlembert equation (implicit differentialequation on Euclidean space TE). In this connection, the latter will be shownto be equivalent to a Lagrange equation (implicit differential equation onmanifold T Q ), which will naturally be obtained and thoroughly discussed.

2.1 Integrable part of dAlembert equationAs to the integration of DdAl , the first step is to extract its integrable part,i.e. the region

D(i)dAl DdAlswept by the tangent lifts of all its integral curves (i.e. covered by the orbits

of such lifts). As to the extraction of D(i)dAl , it is quite natural to start fromthe following remark.

Owing to Prop.1, the base integral curves (if any) of DdAl are constrainedto live in Q (see condition (

)) and then their tangent lifts, i.e. the integral

curves, live in T Q . As a consequence, the tangent lifts of the integral curveslive in T T Q , that is to say,

D(i)dAl T T Q

So we obtain

D(i)dAl DdAl T T Q

8

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Restriction of dAlembert equation

The above result suggests focusing on the restriction of DdAl obtained viaintersection with T T Q , i.e. on the first-order differential equation in implicitform on T Q (second-order on Q )

DLagr := DdAl T T Qwhich will be called Lagrange equation.

Note that Lagrange equation is an effective restriction of dAlembert equa-tion (i.e. DLagr DdAl ) in presence, and only in presence, of two-sidedconstraints (when dim Q < dim

E), as is shown by the following proposition.

Proposition 2 DLagr DdAl , iff dim Q < dim E.

Proof Let dim Q < dim E (whence TpQ E for all p Q ). We shall provethat, under the above hypothesis, DLagr DdAl . Indeed, we can choosep Q , v E TpQ (whence (p, v) T Q ), u = v and w = 1m(F(p, v) +) with Tp Q (whence F(p, v) mw Tp Q ) and clearly we obtain(p, v; u, w) DdAl , (p, v; u, w) T T Q and then (p, v; u, w) DLagr .Conversely, let dim Q = dim E (whence TpQ = T2(p,v)Q = E for all (p, v) T Q ). We shall prove that, under the above hypothesis, DdAl DLagr .Indeed, for any (p, v; u, w)

DdAl , we have p

Q , v

E = TpQ ,

u = v and w E = T2(p,v)Q , that is, (p, v; u, w) T2Q T T Q andthen (p, v; u, w) DLagr .

Extraction of the integrable part

By extracting DLagr from DdAl , via intersection of the latter with T T Q , wejust obtain D(i)dAl = , as is shown by the following proposition.

Proposition 3 D(i)dAl = DLagr =

Proof First we remark that DdAl and DLagr are equivalent equations i.e.they have the same integral curves since, owing to inclusions D(i)dAl DLagr DdAl , condition Im c DdAl (i.e. Im c D(i)dAl ) implies Im c DLagr and,conversely, Im c DLagr implies Im c DdAl .That amounts to saying D(i)dAl = D(i)Lagr .Then we anticipate that DLagr is integrable, i.e. = DLagr = D(i)Lagr .Hence our claim.

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10 2 From dAlembert to Lagrange

So the focal point is now to prove the integrability of DLagr . That willfollow from the stronger property of DLagr being reducible to normal form, aswill be shown in the sequel.

2.2 Lagrange equation

In order to prove the reducibility of Lagrange equation to normal form, weshall need to give a deeper insight into its algebraic formulation and the un-derlying geometric structures. Its integral curves will then be given a globalcharacterization in terms of the above geometric structures (the traditional

local characterization in coordinate formalism will finally be deduced).

Covector formulation

From the set-theoretical point of view, Lagrange equation can be expressed inthe form

DLagr = {(p, v; u, w) T T Q | u = v , m wp = F(p, v)p , p TpQ } T2QFrom the algebraic point of view, the condition on virtual works charac-

terizing DLagr can be given the form of a covector equality, as will now beshown.

Associated with the inertial mass distribution m , there is a semi-basic1-form

[m] : T2Q TQ : (p, v; v, w) (p, [m](p, v, w))on T2Q , called covector inertial field, whose value

[m](p, v, w) := (m w) |TpQ TpQat any (p, v; v, w) T2Q is the virtual work of m w . 1

Associated with the force field F , there is a semi-basic 1-form

F : T Q

TQ : (p, v)

(p, F(p, v))

on T Q , called covector force field, whose value

F(p, v) := F(p, v) |TpQ TpQat any (p, v) T Q is the virtual work of F(p, v) . 2

1 For any u E, we define u E by putting u : v E u v R . Then, for anyp Q , the restriction of u to TpQ yields u |TpQ TpQ .

2 The virtual work of a positional force field f can be regarded as an ordinary 1-form onQ , namely f : p Q (p, f(p)) TQ , f(p) := f(p) |TpQ TpQ .

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Proposition 4 Lagrange equation can be given the covector formulation

DLagr = {(p, v; u, w) T T Q | u = v , [m](p, v, w) = F(p, v)}

Proof Just notice that condition

m w p = F(p, v) p , p TpQ

means

(m w) |TpQ = F(p, v) |TpQ

that is, [m](p, v, w) = F(p, v)

Riemannian geodesic curvature field

The reducibility of DLagr to normal form requires that, for any choice of thedata (p, v) T Q , the algebraic equation [m](p, v, w) = F(p, v) should beuniquely solvable with respect to the unknown w T2(p,v)Q . As the latteronly appears in the left hand side of the equation, the above property is to be

checked through a thorough investigation of [m] .The following geometric considerations showing that such a semi-basic

1-form on T2Q is the transformed of a suitable semi-basic 1-form on T Qthrough a distinguished semi-spray will prove to be crucial.

Remark that [m] is the semi-basic 1-form induced on T2Q by the Eu-clidean metric

gm : E E : u gm(u) := (m u) (positive definite, symmetric, linear map), since its value at any (p, v; v, w) T2Q is

[m](p, v, w) = gm(w)TpQ

TpQIn the same way, there is a semi-basic 1-form

g : T Q TQ : (p, v) (p, gp(v))

induced on T Q by gm , whose value at any (p, v) T Q is

gp(v) := gm(v)TpQ

TpQ

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g is a Riemannian metric on Q, characterized by the quadratic form orLagrangian function

K : T Q R : (p, v) K(p, v) := 12

gp(v) | v = 12

m v v

(the kinetic energy of the mechanical system).Riemannian manifold (Q, K) carries a distinguished semi-spray

K : T Q T2Q : (p, v) (p, v; v, K(p, v))called Riemannian spray, uniquely determined by the following property.

Proposition 5 There exists one, and only one, semi-spray K on T Q s.t.,for all (p, v) T Q ,

gm(K(p, v))TpQ

= 0

Proof (i) Unicity First remark that, for any (p, v) T Q , the map

w T2(p,v)Q() gm(w)

TpQ

TpQ

is injective, since

gm(w1)TpQ

= gm(w2)TpQ

with w1, w2 T2(p,v)Q and then w1 w2 TpQ implies

gp(w1 w2) = gm(w1 w2)TpQ

= gm(w1)TpQ

gm(w2)TpQ

= 0

that is, recalling that gp : TpQ TpQ is a linear isomorphism,w1 = w2

So, if there exists a vector in T2(p,v)Q whose image through () is zero, it isunique.

(ii) Existence For any (p, v) T Q , putK(p, v) := w + u T2(p,v)Q

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with w T2(p,v)Q , u := g1p

gm(w)TpQ

TpQ

The image of K(p, v) through () is zero, since

gm(K(p, v))TpQ

= gm(w + u)TpQ

= gm(w)TpQ

+ gm(u)TpQ

= gm(w)TpQ

+ gp(u)

= gm(w)TpQ

gm(w)TpQ

= 0

K transforms g (semi-basic 1-form on T Q ) into

[K] : T2Q TQ : (p, v; v, w) (p, [K](p, v, w))(semi-basic 1-form on T2Q ) by putting, for any (p, v; v, w) T2Q ,

[K](p, v, w) := gp(w K(p, v)) TpQ[K] will be called Riemannian geodesic curvature field.

Actually [K] does not differ from [m] , as is shown in the following propo-sition.

Proposition 6 Lagrange equation, in covector formulation, reads

DLagr = {(p, v; u, w) T T Q | u = v , [K](p, v, w) = F(p, v)}

Proof Owing to Prop. 4, it will suffice to show that

[K] = [m]

To this end, note that, for any (p, v; v, w) T2Q , from Prop. 5 we obtain

[K](p, v, w) := gp(w K(p, v))= gm(w K(p, v))

TpQ

= gm(w)TpQ

gm(K(p, v))TpQ

= gm(w)TpQ

= [m](p, v, w)

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Normal form

The reducibility of Lagrange equation to normal form immediately follows fromProp. 6.

Consider the vertical force field

F := g1 F : T Q T Q : (p, v) (p, F(p, v))

F(p, v) := g1p (F(p, v)) TpQ

and the semi-spray

:= K + F : T Q

T2Q : (p, v)

(p, v; v, (p, v))

(p, v) := K(p, v) + F(p, v) T2(p,v)Q

Proposition 7 Lagrange equation can be put in the normal form

DLagr = Im = {(p, v; u, w) T T Q | u = v , w = (p, v)}

Proof Just notice that covector equality

[K](p, v, w) = F(p, v)

readsgp(w K(p, v)) = F(p, v)

w K(p, v) = g1p (F(p, v))w = K(p, v)) + F(p, v)

w = (p, v)

Integral curves

The condition characterizing the integral curves of Lagrange equation can now

be formulated as follows.

Letc : I T Q : t c(t) = (p(t), v(t))

be a smooth curve of T Q and

Q c : I Q : t (Q c)(t) = p(t)its projection onto Q . 3

3 The tangent lift of Q c will be denoted by (Q c) .

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Proposition 8 c is an integral curve of DLagr , iff(Q c) = c , [K] c = F c ()

or, in normal form,c = c ()

Proof As is known, c is an integral curve of DLagr , iffIm c DLagr

that is to say, for all t I,() c(t) = (p(t), v(t); p(t), v(t)) DLagr

(i) Owing to Prop. 6, condition () readsp(t) = v(t) , [K](p(t), v(t), v(t)) = F(p(t), v(t))

that is,

(Q c)(t) = (p(t), p(t))= (p(t), v(t))

= c(t)

and

([K] c)(t) = ( p (t), [K](p(t), v(t), v(t)) )= ( p(t), F(p(t), v(t)) )

= (F c)(t)That proves our first claim.

(ii) Owing to Prop. 7, condition () also reads

p(t) = v(t) , v(t) = (p(t), v(t))

that is,

c(t) = (p(t), v(t); p(t), v(t))

= (p(t), v(t); v(t), (p(t), v(t)))

= ( c)(t)That proves our second claim.

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As a consequence, the condition characterizing the base integral curves ofLagrange equation will be formulated as follows.

Let : I Q : t (t) = p(t)

be a smooth curve of Q and

[K] : I TQ

its Riemannian geodesic curvature, whose vector (rather than covector) ex-pression is the covariant derivative 4

dt

:= g1 [K] : I T Q

Proposition 9 is a base integral curve of DLagr , iff

[K] = F ()

or, in vector formulation,dt

= F

or, in normal form, = ()

Proof Recall that a base integral curve of Lagrange equation is the projection = Q c of an integral curve c (smooth curve of T Q satisfying condition() or the equivalent ()).

(i) Now, if is a base integral curve, condition () implies = (Q c) = cwhence = c and then (). Conversely, if satisfies condition (), then itis obviously a base integral curve (projection of c := satisfying ()). Clearly,condition (

) is equivalent to

g1 [K] = g1 F

that is,dt

= F which is the above mentioned vector formulation of ().

4 Covariant derivative is also related to an important geometric structure, called Levi-Civita connection of Riemannian manifold (Q, K) .

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(ii) In the same way, through condition (), one can show that the base integralcurves are characterized by ().Alternatively, from

dt

: t I (p(t), p(t); p(t), p(t)) T2Q[K]

p(t), gp(t) (p(t) K(p(t), p(t)))

TQ

g1

p(t), p(t) K(p(t), p(t))

T Q

and

K : t I (p(t), p(t) K(p(t).p(t))) T Qthat is,

dt

= K we deduce that the vector formulation of () (which has already been seen tocharacterize the base integral curves) is equivalent to

= K + F

which is condition (

).

From the dynamical point of view, some remarks are now in order.

For F = 0 , the base integral curves characterized by a vanishing Rie-mannian geodesic curvature [K] = 0 coincide with the inertial motionsof S (i.e. the motions which would be possible if F were zero).

The effect of a covector force field F = 0 is then that of deviating theDPMs of S from the inertial trend, by giving them a non-vanishing Rieman-nian geodesic curvature, namely [K] = F .

Classical Lagrange equations

The scalar equations obtained with the aid of a chart by orderly equalling thecomponents of the covector or vector-valued functions which appear in the leftand right hand sides of () or (), are the classical Lagrange equations ofAnalytical Dynamics.

They will prove to be only locally equivalent to the geometric Lagrangeequation, in the sense that they only characterize the base integral curves ofthe latter which live in the coordinate domain of the given chart.

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Preliminaries

Consider the coordinate domain U= Im of a chart : q W (q) U,expressing the points p U Q in function of coordinates q W Rn(with n := dim Q ). 5

To any smooth curve : t I (t) = p(t) Q living in U, i.e.satisfying

p = p(t) Ufor all t I, there corresponds in a smooth coordinate expression

q = q(t) Wrelated to by p(t) = (q(t)) , also denoted

p = (q) (1)

(dependence from time t is understood).

To the first tangent lift c = , that is,

p = p(t) U, v = v(t) Tp(t)Qwith

v = p

there corresponds in a smooth coordinate expression

q = q(t) W , v = v(t) Rn

related to by (1) and the time derivative of (1) 6

v = v(q, v) := vhp

qh

q

(2)

wherev = q

is the n-tuple of linear components of v in . Remark that, for all h =1, . . . , n ,

v

vh

(q,v)

=p

qh

q

,v

qh

(q,v)

= vk2p

qhqk

q

=d

dt

p

qh

q

(3)

5 Recall that, for any q = 1(p) W , the partial derivatives

pq1

q

, . . . , pqn

q

provide a basis of TpQ .6 A repeated index, in upper and lower position, denotes summation over (1, . . . , n).

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To the second tangent lift c = , that is,

p = p(t) U, v = v(t) Tp(t)Q , w = w(t) T2(p(t),v(t))Q

with

v = p , w = v = p

there corresponds in a smooth coordinate expression

q = q(t) W , v = v(t) Rn , w = w(t) Rn

related to by (1), (2) and the time derivative of (2)

w = w(q ,v ,w) := whp

qh

q

+ vhvk2p

qhqk

q

where

w = v = q

is the n-tuple of affine components of w in .

In the sequel, the components of F in will be denoted by

Fh(q, v) := (F )h = (F(p, v))h =

F(p, v) | pqh

q

= F(p, v) p

qh

q

and the components of F = g1 F will be denoted by

Fi(q, v) := (F )i = (F(p, v))i = (g1p (F(p, v)))i = gih(q) (F(p, v))h = gih(q) Fh(q, v)

where [ghk(q)] is the inverse of the nonsingular matrix [ghk(q)] defined by

ghk(q) :=

gp

p

qh q |p

qk q

= mp

qh q p

qk q

Lagrange equations

The above coordinate formalism will now be adopted for the characterizationof the base integral curves of DLagr living in the given coordinate domain.

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Proposition 10 A smooth curve : t I p = (q(t)) U Q livingin the coordinate domain U of a chart of Q is a base integral curve ofDLagr , iff its coordinate expression q = q(t) satisfies (for all h, i = 1, . . . , n )the classical Lagrange equations

q = v ,d

dt

K

vh

(q,v)

Kqh

(q,v)

= Fh(q, v) ()h

or, in normal form, 7

q = v , vi =

i

jk

q

vjvk + Fi(q, v) ()i

Proof (i) Recall that is a base integral curve of DLagr , iff it satisfies equation(). As lives in the coordinate domain of , equation () is equivalent tothe n scalar equations obtained by orderly equalling the components in ofits left and right hand sides, i.e.

([K] )h = (F )h ()hThe components Fh(q, v) := (F )h have already been shown in the abovepreliminaries, where we have put v = q. The components ([K] )h will nowbe evaluated. To this end, by making use of (3) and usual rules of derivation,

we obtain

([K] )h := ([K](p, v, w))h=

[K](p, v, w) | p

qh

q

= mw(q ,v ,w) pqh

q

=d

dt

mv(q, v) p

qh

q

mv(q, v) d

dt

p

qh

q

=

d

dt

mv

(q, v) v

vh(q,v)

mv(q, v) v

qh(q,v)

7 Here we encounter the Christhoffel symbols (of the Levi-Civita connection) associatedwith a Riemannian manifold, defined on the coordinate domain of any chart by

i

jk

:=

1

2gih(jgkh + kghj hgjk)

with

ighk :=ghk

qi

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=

d

dt

vh(q,v)

12mv v

qh(q,v)

12mv v

=d

dt

K

vh

(q,v)

Kqh

(q,v)

So equations ()h just take the form ()h.(ii) Also recall that is a base integral curve of DLagr , iff it satisfies equation(). As lives in the coordinate domain of , equation () is equivalent tothe n scalar equations obtained by orderly equalling the affine components in of its left and right hand sides, i.e.

i

= ( )i

() i

where

i = wi

= vi

and

( )i = (K + F )i= (K )i + (F )i= iK(q, v) + F

i(q, v)

The components Fi(q, v) := (F )i have already been shown in the abovepreliminaries. The components iK(q, v) := (K )i will now be evaluated.To that purpose we need the coordinate expression of K, which is given by

K(p, v) =1

2

gp

vh

p

qh

q

| vk p

qk

q

=1

2ghk(q) v

hvk

whence

K

qh(q,v) =

1

2(hgjk)q vj

vk

K

vh

(q,v)

= ghk(q)vk

and

2K

vhqk

(q,v)

= (kghj)q vj

2K

vhvk

(q,v)

= ghk(q)

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As a consequence, the above components ([K] )h can more explicitly beexpressed in the form

([K] )h = ([K](p, v, w))h=

2K

vhvk

(q,v)

wk +2K

vhqk

(q,v)

vk Kqh

(q,v)

= ghk(q)wk + (kghj)q v

jvk 12

(hgjk)q vjvk

= ghk(q)wk +

1

2(kghj)q v

jvk +1

2(jghk)q v

jvk 12

(hgjk)q vjvk

= ghk(q)wk

+

1

2(kghj + jgkh hgjk)q vj

vk

whence

dt

i= (g1 ([K] ))i

= (g1p ([K](p, v, w)))i

= gih(q) ([K](p, v, w))h

= wi +

i

jk

q

vjvk

Therefore, identities 8

dt

i= ( K )i

= i (K )i

read

wi +

i

jk

q

vjvk = wi iK(q, v)

Hence we obtain

iK(q, v) =

i

jk

q

vjvk

So equations () i just take the form ()i.

8 See the proof, part (ii), of Prop. 9.

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2.3 Euler-Lagrange equationA special mention, for its primary role in both mathematical and theoreticalphysics, is to be given to the dynamics of a conservative system.

Conservative system

Such a name refers to a system S= (Q,m, f) carrying a conservative field f ,i.e. a positional force field whose virtual work

f = dV

is an exact 1-form, deriving from a smooth potential energyV : Q R

(determined up to a locally constant function).The name of conservative is due to the following conservation law of

mechanical energy

E := K+ V : T Q R : (p, v) E(p, v) := K(p, v) + V(p)(kinetic energy plus potential energy).

Proposition 11 Along each integral curve c of

DLagr = {(p, v; u, w) T T Q | u = v , [K](p, v, w) = dpV}the mechanical energyE keeps constant, i.e.

E c = const.Proof If c = (p, v) denotes an integral curve and w := v , from p = v and[K](p, v, w) = dpV it follows that

d

dt(E c) = d

dtE(p, v)

=d

dtK(p, v) +

d

dtV(p)

= ddt( 12m v v) + ddtV(p)= m w v + d

dtV(p)

= [K](p, v, w) | v + dpV | v= dpV | v + dpV | v= 0

Hence our claim.

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Lagrangian geodesic curvature field

In conservative dynamics, the kinetic energy and the potential energy whichare the two ingredients generating DLagr can be merged into a unique object,as follows.

Define a new Lagrangian function by putting

L := K V : T Q R : (p, v) L(p, v) := K(p, v) V(p)(kinetic energy minus potential energy).

Associated with L , there is a Lagrangian geodesic curvature field given by

[L] := [K] + dV : T2Q

TQ : (p, v; v, w)

(p, [L](p, v, w))

with[L](p, v, w) := [K](p, v, w) + dpV TpQ

(Riemannian geodesic curvature field minus conservative field).From the above definitions, it follows that

DLagr = DEulLagrthat is, DLagr does not differ from the Euler-Lagrange equation

DEulLagr := {(p, v; u, w) T T Q | u = v , [L](p, v, w) = 0}

generated by L .

Integral curves

The conditions characterizing the integral curves and the base integral curvesof Euler-Lagrange equation can now be formulated as follows.

Proposition 12 A smooth curve c of T Q is an integral curve of DEulLagr ,iff

(Q c) = c , [L] c = 0 ()Proof The same as the proof, part (i), of Prop.8.

Proposition 13 A smooth curve of Q is a base integral curve ofDEulLagr ,iff its Lagrangian geodesic curvature vanishes, i.e.

[L] = 0 ()

Proof The same as the proof, part (i), of Prop.9.

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Classical Euler-Lagrange equations

The local scalar equations obtained with the aid of a chart by equalling tozero the components of the covector-valued function which appears in the lefthand side of (), are the classical Euler-Lagrange equations of AnalyticalDynamics.

Proposition 14 A smooth curve : t I p = (q(t)) U Q livingin the coordinate domain U of a chart of Q is a base integral curve ofDEulLagr , iff its coordinate expression q = q(t) satisfies (for all h = 1, . . . , n )the classical Euler-Lagrange equations

q = v ,d

dt

L

vh

(q,v)

Lqh

(q,v)

= 0 ()h

Proof Recall that is a base integral curve of DEulLagr , iff it satisfies equa-tion (). As lives in the coordinate domain of , equation () is equivalentto the n scalar equations obtained by equalling to zero the components in of its left hand side, i.e.

([L] )h = 0 ()hThe above components are given by

([L] )h := ([L](p, v, w))h=

[L](p, v, w) | p

qh

q

=

[K](p, v, w) | p

qh

q

+

dpV | p

qh

q

=d

dt

K

vh

(q,v)

Kqh

(q,v)

+V

qh

q

=

d

dt

(K

V)

vh(q,v)

(K

V)

qh(q,v)

=d

dt

L

vh

(q,v)

Lqh

(q,v)

where we have put v = q.

So equations ()h just take the form ()h.

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Remark For both L := K and L := K V , the semi-basic 1-form [L] :T2Q TQ is a mapping which takes any (p, v; v, w) T2Q to the covector(p, [L](p, v, w)) TQ characterized, in a chart, by components

([L](p, v, w))h =2L

vhvk

(q,v)

wk +2L

vhqk

(q,v)

vk Lqh

(q,v)

Such a coordinate technique can be used to define [L] for an arbitrary La-grangian function L : S T Q R (smooth on an open subset S of T Q ),since also in such a case the value [L](p, v, w) (for any (p, v) S), definedby the above components ([L](p, v, w))h in a given chart, turns out to be an

invariant covector.9

2.4 Variational principle of stationary action

Euler-Lagrange equation takes conservative dynamics into the classical area ofvariational principles, as will now be shown.

Variational calculus

Let

: I Q : t p(t)be a smooth curve of Q .

A smooth variaton of with fixed end-points in a closed sub-interval of I,is a smooth mapping

: (, ) J Q : (s, t) p(s, t)

(with > 0 and J I) satisfying

p(0, t) = p(t) , t J

and, at the end-points of a closed interval [t1, t2] J,p(s, t1) = p(t1) , p(s, t2) = p(t2) , s (, )

can be thought of as a one-parameter family

{ s : J Q : t ps(t) := p(s, t) }s(,)9 Actually, through higher geometric methods, [L] can be given a coordinate-free defi-

nition.

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of varied curves near |J = o with fixed end-points in [t1, t2] J I,whose tangent lifts { s }s(,) are all included in

: (, ) J T Q : (s, t)

p(s, t),p

t

(s,t)

also define a one-parameter family

{ t : (, ) Q : s pt(s) := p(s, t) }tJ

of isocronous curves, whose tangent vectors at the points of

|J are the values

ofo : J T Q : t

p(t),

p

s

(0,t)

Now consider, in a closed sub-interval [t1, t2] of I, the action of , i.e.the integral 10

I :=t2t1

(L ) dt

For any smooth variation of with fixed end-points in [t1, t2], theaction of is then the real-valued function

I :=t2t1

(L ) dt : (, ) R : s Is :=t2t1

(L s) dt

At s = 0 , the value of I is I and its derivative

I := dIds

0

is called the first variation of I at .

Proposition 15 For every smooth variation of with fixed end-points ina closed sub-interval [t1, t2] of I, the first variation of I at is related tothe Lagrangian geodesic curvature of by

I = t2t1

[L] | o dt

10 All of the following considerations and results of variational theory, hold true for anarbitrary smooth Lagrangian function as well (on this matter, also recall the final Remarkof the previous section).

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Proof First remark that

I := dds

0

t2t1

(L ) dt =t2t1

(L )s

s=0

dt

Now, for any given to J, consider a chart on a neighbourhood U of thepoint p(to) Im . In a suitably small neighbourhood of (0, to) (, )J, takes values in U (by continuity) and will then have a local coordinate ex-pression (qh(s, t)). As a consequence, will have local coordinate expression

qh(s, t), qh

t

(s,t)

, denoted, for s = 0, by (q = q(t), v = v(t)) with v = q,

and o will have components oh := q

h

s s=0

. With the aid of , we obtain

(L )s

(0,to)

=L

qh

(q(to),v(to))

qh

s

(0,to)

+L

vh

(q(to),v(to))

2qh

st

(0,to)

=L

qh

(q(to),v(to))

qh

s

(0,to)

+

d

dt

to

L

vh

(q,v)

qh

s

s=0

d

dt

to

L

vh

(q,v)

qh

s

(0,to)

=

L

qh

(q(to),v(to))

ddt

to

L

vh

(q,v)

o

h(to) +d

dt

to

L

vh

(q,v)

oh

that is, 11

(L )s

(0,to)

= [L] | o(to) +d

dt

toFL | o

So, on the whole J, we have

(L )s

s=0

= [L] | o +d

dtFL | o

As o(t1) = 0 and o(t2) = 0 , we also have

t2t1

ddt

FL | o dt = FL | o(t2) FL | o(t1) = 0

Hence our claim.

11 FL : I TQ will denote the (invariant) covector-valued map whose componentsin are given by

(FL )h = Lvh

(q,v)

(see the final Remark of Legendre transformation in Chap. 3).

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Variational principle

A smooth curve : I Q is said to be a geodesic curve of (Q,L) , if it satisfiesHamiltons variational principle of stationary action, owing to which, for everysmooth variation of with fixed end-points in a closed sub-interval of I,

I is required to be stationary at , that is to say,I = 0 ,

The above variational principle is completely equivalent to Euler-Lagrangeequation.

Proposition 16 The geodesic curves of (Q,L) are the base integral curves ofDEulLagr .

Proof (i) If : I Q is a base integral curve of DEulLagr , i.e.[L] = 0

we clearly obtain, for every smooth variation of with fixed end-points ina closed sub-interval [t1, t2] of I,

t2

t1 [L]

|o

dt = 0

and hence, owing to Prop.15, is a geodesic curve of (Q,L) .

(ii) Conversely, if : t I p(t) Q is not a base integral curve ofDEulLagr , say

([L] )(to) = 0 , to Iwe shall prove that, for a suitable smooth variation of with fixed end-points in a closed sub-interval [t1, t2] of I,

t2

t1 [L]

|o

dt

= 0

which, owing to Prop.15, means that is not a geodesic curve of (Q,L) .

To this end, consider a chart on a neighbourhood

U p(to)Owing to our hypothesis, at least one of the components ([L] )h is non-nullat to , say

([L] )1(to) > 0

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By continuity, there exists a suitably small open interval J I containing tos.t., for all t J,p(t) U, ([L] )1(t) > 0

Now, for each s (, ) with a suitably small > 0 and each t J, weconsider the point p(s, t) U of coordinates

q1(s, t) := q1(t) + s(cos(t to) cos)q2(s, t) := q2(t)

. . . . . .

qn(s, t) := qn(t)

where (qh(t)) is the n-tuple of coordinates of p(t) and

0 < 0

Hence

t2t1

[L] | o dt =t2t1

([L] )h o h dt =t2t1

([L] )1 o 1 dt > 0

which is our claim.

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Riemannian case

For L = K, the above variational theory characterizes the geodesic curves ofRiemannian manifold (Q, K) through condition

[K] = 0

Owing to the positive definiteness of K, from the conservation law ofenergy E = K it follows that, on the one hand, a geodesic curve satisfying

K = const. = 0

degenerates into a singleton and, on the other hand, a non-degenerate geodesiccurve is a uniform motion, that is,

K = const. > 0

The geometric meaning of non-degenerate geodesic curves will now beshown.

Let us consider one more Lagrangian function, namely

:=

2K

which is clearly smooth on T Q \ K1(0). 12

If : t I Q : t p(t) Q is a smooth curve satisfying

Im T Q \ K1(0)

(i.e. p(t) = 0 for all t I) and [t1, t2] I, the integral

L :=t2t1

( ) dt

defines the length of the arc of with end-points (p(t1), p(t2)) .13

12 is smooth only on T Q \ K1(0) , since, in any chart, its partial derivatives

qh

(q,v)

=1

2K(q, v)

K

qh

(q,v)

,

vh

(q,v)

=1

2K(q, v)

K

vh

(q,v)

are not even defined for v = 0 , i.e. for v = 0 or, equivalently, (p,v) K1(0).13 For any dt > 0 , (p(t), p(t)) dt =

gp(t)(dp) | dp > 0 defines the length, in thegiven Riemannian metric, of the infinitesimal arc dp := p(t)dt .

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The variational theory previously sketched, applied to ,14

shows that is a curve of stationary length, that is,

L = 0 ,

iff[] = 0

Proposition 17 is a non-degenerate geodesic curve of (Q, K) , iff it is auniform motion of stationary length.

Proof The above result is an immediate consequence of the followingLemma If is a uniform motion, that is,

K = = const. > 0

then

[] = 12

[K]

In order to prove the lemma, recall that, along the uniform motion : t I p(t) Q , we have (for any given t I and any chart at p(t) ) 15

([] )h = ddt

vh

(q,v)

qh

(q,v)

=d

dt

12

K

vh

(q,v)

1

2

K

qh

(q,v)

=12

d

dt

K

vh

(q,v)

Kqh

(q,v)

=12

([K] )h

which is our claim.

14 See footnote 10.15 See footnote 12.

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Chapter 3

From Lagrange to Hamilton

The Lagrangian dynamics of a conservative system S whose geometricalarena is the velocity phase space T Q , supporting Euler-Lagrange equation will now be taken into the range of Hamiltonian dynamics, where the geo-metrical arena is the momentum phase space TQ , supporting a Hamiltonequation which, defined in terms of the canonical symplectic structure of TQ,directly arises in normal form.

3.1 Legendre transformation

The classical transition from velocity to momentum phase space is providedby the well known map which takes any velocity (p, v) T Q onto the corre-sponding momentum (p, m v |TpQ) TQ .

Lagrangian function and Legendre transformation

The above map is indeed the vector bundle isomorphism

g : T Q TQ : (p, v) (p, gp(v)) , gp(v) := m v |TpQowing to which the configuration space of the system has been given the struc-

ture of a Riemannian manifold (Q, K) . 1Such an isomorphism is also said to be the Legendre transformation de-

termined by Lagrangian function L = K V (in terms of which it can beexpressed) 2 and is denoted by

FL : T Q TQ : (p, v) (p, FpL(v)) , FpL(v) := gp(v)(whence Q FL = Q ).

1 See Riemannian geodesic curvature field in Chap. 2.2 See the next Coordinate formalism.

33

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34 3 From Lagrange to Hamilton

Coordinate formalism

Recall that TQ is a 2n-dimensional manifold, where each element (p, ) iscompletely characterized, in a natural chart, by two n-tuples of coordinates(q, p) , namely the coordinates q = (q1, . . . , q n) of p = (q), in a chart of

Q , and the components p = (p1, . . . , pn) of in , given by ph = | pqhq .

In natural coordinate formalism, Legendre transformation

(p, v) (p, ) , = FpL(v)

is expressed by 3

(q, v) (q, p) , ph = ghk(q) vk = Lvh

(q,v)

(1)

The inverse transformation

(p, ) (p, v) , v = (FpL)1()

is in turn expressed by

(q, p) (q, v) , vh = ghk(q)pk =: h(q, p) (2)

Remark Such a coordinate technique can be used to define a Legendremorphism FL for an arbitrary Lagrangian function L : S T Q TQ(smooth on an open subset S of T Q ), since also in such case the value = FpL(v) TpQ (for any (p, v) S), defined by the above components

ph =L

vh

(q,v)

in a given chart, turns out to be an invariant covector. 4

L is then said to be a regular or a singular Lagrangian function, accordingto whether FL is injective or not.

3 Recall that

ph =

gp(v) | pqh

q

=

gp p

qh

q

| v = g(q) pqh

q

| pqk

q

vk = ghk(q) v

k =K

vh

(q,v)

=L

vh

(q,v)

4 Actually, through higher geometric methods, FL can be given a coordinate-free def-inition.

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3.2 Hamiltonian functionLegendre transformation also determines transition from the Lagrangian func-tion (on T Q ) to a Hamiltonian function (on TQ ) .

Energy and Hamiltonian function

Recall that, associated with Lagrangian function L = K V , there is theenergy function 5

E := K+ V = 2K

L =

FL

|idTQ

L : T Q

R

The Hamiltonian function

H := E (FL)1 : TQ R

is the push forward of E by Legendre transformation FL .


In natural coordinate formalism, the Hamiltonian function

(p, ) H(p, ) = E((FL)1(p, )) = E(p, (FpL)1()) = E(p, v) = FpL(v) | v L(p, v= | v L(p, v) , v = (FpL)1()

is expressed by

(q, p) H(q, p) = pk vk L(q, v) , v = (q, p)

Hence, owing to (1) and (2), we obtain

H

qh(q,p)

= pkvk

qh(q,p)

L

vk(q,v)

vk

qh(q,p)

L

qh(q,v)

=

L

qh(q,v)

(3)

andH

ph

(q,p)

= vh +pkvk

ph

(q,p)

Lvk

(q,v)

vk

ph

(q,p)

= h(q, p) (4)

5 We put

FL | idTQ : (p,v) T Q FpL(v) | idTQ(v) = FpL(v) | v = gp(v) | v = 2K(p,v) R

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3.3 Hamiltonian vector fieldThe canonical symplectic geometry of TQ will now be taken into considera-tion.

Canonical symplectic structure and Hamiltonian vector field

Let : T M TM : (, X) (, (X))

be the canonical symplectic structure of cotangent bundle M := TQ . 6

As is a vector bundle isomorphism of T M onto TM, it transforms

the differentialdH : M TM

of Hamiltonian function H into the Hamiltonian vector field

XH := 1 dH : M T M


Recall that, on the 2n-dimensional manifold M = TQ , the canonical sym-plectic structure is characterized, in any natural chart, by the 2n 2nmatrix of components

(1)(1) (1)(2)(2)(1) (2)(2)

with (1)(1) = (2)(2) = 0 (nn zero matrix) and (1)(2) = (2)(1) = (nnidentity matrix).

If, for any M, dH(1) dH(2)

and

XH()(1)

XH()(2)

are the 2n-tuples of components of dH TM and XH() TM, respec-tively, the linear mapping

dH = (XH())

is expressed by 7

dH(1) = ()(1)XH()() , dH(2) = ()(2)XH()

()

6 Any point (p, ) TQ is now referred to as M.7 Summation over () = (1), (2) is understood.

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that is, dH(1) = XH()(2) , dH(2) = XH()(1)As

dH(1) =

H

qh

(q,p)

, dH(2) =

H

ph

(q,p)

we obtain

XH()(1) =

H

ph

(q,p)

, XH()

(2) =

H

qh

(q,p)

(5)

3.4 Hamilton equation

Remark that Legendre transformation FL : T Q TQ transforms eachsmooth curve c : I T Q of velocity phase space into a smooth curve FLc :I TQ of momentum phase space.

Cotangent dynamically possible motions

If c is an integral curve of DEulLagr , i.e. a TDPM of S, then FL c willbe said to be a cotangent dynamically possible motion (CDPM) of S.

As FL is an isomorphism, TDPMs and CDPMs bijectively correspond

to one another. Through such a bijection, the problem of determining theTDPMs proves to be naturally equivalent to that of determining the CDPMs.

Hamilton equation

Proposition 18 The CDPMs are the integral curves of Hamilton equation

DHam := Im XH

Proof Letk : I TQ , c : I T Q

be smooth curves corresponding to each other through Legendre transforma-tion, i.e.

k = FL cand then both projecting down onto the same curve

Q k = Q FL c = Q c : I QWe shall prove that

() (Q c) = c , [L] c = 0 k = XH k ()

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For any t I, the natural coordinates (q, p) = (q(t), p(t)) of k = k(t) arethen related to the natural coordinates (q, v) = (q(t), v(t)) of c = c(t) by (1)and (2), i.e.

ph =L

vh

(q,v)

, vh = h(q, p).

So condition (), that is,

qh = vh ,d

dt

L

vh

(q,v)

Lqh

(q,v)

= 0

is equivalent, owing to (1) and (2), to

qh = h(q, p) , ph = Lqh

(q,v)

and then, owing to (3) and (4), to

qh =H

ph

(q,p)

, ph = Hqh

(q(p)

()h

which, owing to (5), is just condition (). 8

Remark the second-order character of the above Hamilton equation, ex-hibeted by the fact that any integral curve k of DHam is the Legendre lift ofits own projection Q k , i.e.

k = FL c = FL (Q c) = FL (Q k)

The integral curves and the base integral curves of DHam are then bi-jectively related to one another by projection Q and, owing to the abovebijection, the first-order problem of determining the integral curves of DHamproves to be naturally equivalent to the second-order problem of determiningits base integral curves, i.e. the solution curves of

(FL ) = XH (FL )

Classical Hamilton equations

The local scalar equations ()h, obtained in a natural chart by orderly equallingthe components of the left and right hand side of (), are the classical Hamil-ton equations of Analytical Dynamics.

8 Recall thatqh

ph

are the components of k .

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Chapter 4

Concluding remarks

Some comments on the previous results and some perspectives on further de-velopments, now follow.

4.1 Inertia and force

Propositions 1, 3 and 6 have shown that the mathematical model actuallyunderlying classical dynamics is given by the triplet

M:= (Q,K,F)

which will still be called mechanical system.The corresponding Lagrange equation

DLagr = {(p, v; v, w) T T Q | u = v , [K](p, v, w) = F(p, v)}is then the dynamics of M , the DPMs of M being defined as the baseintegral curves of DLagr and hence characterized by condition

[K] = F If F = 0 , the DPMs of

M characterized by a Riemannian geodesic

curvature [K] identically vanishing coincide with the geodesic curves ofRiemannian manifold (Q, K) , which can therefore be called inertial motionsin (Q, K) .

If F = 0 , the DPMs of M characterized by a Riemannian geodesiccurvature [K] differing from zero as much as is imposed by the covectorforce F are then to be called forced motions in (Q, K) , since they appearto be perturbed with respect to the above inertial trend.

So K and F seem to correspond to the empirical notions of inertia(defining inertial motions) and force (defining forced motions), respectively.

39

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40 4 Concluding remarks

4.2 Gauge transformationsHowever the above notions of inertia and force are not uniquely determinedby dynamics.

In fact, there exist gauge transformations of M , which alter the geo-metrical ingredient K and the dynamical ingredient F without altering thedynamics itself, as will now be shown.

For any real-valued smooth function V : Q R , consider the gaugetransformation

K L := K V , F F := F + dVOn the one hand, such a transformation gives rise to a new mechanical

systemM := (Q,L,F)

with Lagrange equation

DLagr = {(p, v; v, w) T2Q | u = v , [L](p, v, w) = F(p, v)}and DPMs the base integral curves of DLagr characterized by condition

[L] = F

If F = 0, the DPMs of M characterized by a Lagrangian geodesiccurvature [L] identically vanishing coincide with the geodesic curves ofLagrangian manifold (Q,L) , which will therefore be called inertial motionsin (Q,L) .

If F = 0, the DPMs of M characterized by a Lagrangian geodesiccurvature [L] differing from zero as much as is imposed by the covectorforce F are then to be called forced motions in (Q,L) , since they appearto be perturbed with respect to the above inertial trend.

On the other hand, from the very definitions of

[L](p, v, w) := [K](p, v, w) + dpV

andF(p, v) := F(p, v) + dpV

for all (p, v; v, w) T2Q , it follows that[L](p, v, w) = F(p, v)

iff[K](p, v, w) = F(p, v)

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4 Concluding remarks 41

whence DLagr = DLagr

So transition from M to M just leads to different specifications of theconventional notions of inertia and force, without altering the observable classof DPMs.

4.3 Geometrizing physical fields

From the physical point of view, the meaning of the above kind of gauge

transformations can be illustrated as follows.Think of V as the potential energy of a conservative component of F ,

that is to say, put F = F + f , f being a conservative field with virtual workf = dV .

Now look at the ingredients L and F of M .On the one hand, F = F f is the virtual work of F = F f , that

is to say, the conservative field does not appear any more in the dynamicalingredient of M .

On the other hand, L embodies V , that is to say, the conservative fieldis encompassed in the geometrical ingredient of M as a potential function,whose effect is that of transforming the natural Riemannian geometry K of

the particle system into a perturbed Lagrangian geometry L = K V .That anticipates, in a sense, Einsteins idea of geometrizing physical fields.

4.4 Geometrical dynamics

From the mathematical point of view, gauge transformations suggest a gener-alized geometrical dynamics.

In such a theory, a mechanical system will be conceived as an arbitrarytriplet M consisting of a smooth manifold Q equipped with a (regular or asingular) Lagrangian function L and a semibasic 1-form F (both smooth on

an open subset of T Q ).The dynamics of M will then be defined by the corresponding Lagrange

equation DLagr (which will or will not be reducible to normal form, accordingto whether L is a regular or a singular Lagrangian function, respectively).

The global analysis of DLagr , its Hamiltonian formulation (or Hamilton-Dirac formulation, in the general case of L being a singular Lagrangian) andfurther generalizations, as well as applications to classical and relativistic dy-namics, are the object of (part of) the current research work on geometricaldynamics.

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Chapter 5

Appendix

The geometry of smooth manifolds embedded in Euclidean affine spaces, isthe main topic of this Appendix. 1 Tangent bundles and differential equa-tions, cotangent bundles and differential forms, will be made to fall into itsrange. The above geometry will finally be re-read, so as to lead to the mod-ern approach to smooth manifolds (which does away with any reference toembeddings into Euclidean environments).

5.1 Submanifolds of a Euclidean space

A submanifold of a Euclidean affine space will be presented as a locus whichlocally exhibits the same differential-topological behaviour as an affine sub-space (that is suggested from the familiar circumstance of smooth curves orsurfaces locally resembling straight lines or planes, respectively).

Affine subspaces

Let E be an m-dimensional Euclidean affine space, modelled on a vector spaceE (acting freely and transitively on E as an Abelian group of operators). 2

A subset A E is said to be an n-dimensional Euclidean affine subspaceof E, if it is the orbit of a point o E under the action of an n-dimensionalvector subspace A E, i.e. A = o + A := {o + v : v A} . 3

1The main (linear, metric, topological) properties of Euclidean affine spaces and thefundamentals of differential calculus on such spaces are the prerequisites for reading thisAppendix.

2 E is meant to be a vector space on the field R of real numbers. The elements of Ewill be called free vectors, whereas for any point p E the elements of {p} E will besaid to be vectors attached at p. The action of E on E will be denoted by + .

3 The name affine subspace is clearly due to the fact that condition A = o + A is

42

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5 Appendix 43

Such a subspace A can be described in E by means of global affineparametrizations (or systems of affine coordinates), sayA = Im

with 4

: Rn E: q p = (q) = o + (q) = o + qheh

(where o A and : v Rn (v) = vheh E is the injective linear mapdetermined by a basis (e1, . . . , en) of A ).

From the differential-topological point of view, the above kind of parametriza-

tion exhibits the following well known properties:

(1) : Rn A is a homeomorphism; 5

(2) : Rn E is C- differentiable, with injective differential dq = at each q Rn. 6

Embedded submanifolds

Let E be an m-dimensional Euclidean affine space (modelled on E).A subset Q E carrying the structure of an n-dimensional embedded

submanifold of E (or smooth manifold embedded in E), is one s.t. each pointp Q admits an open neighbourhood U, in the subspace topology of Q ,which is the image

U= Im of a local parametrization

: W Rn E: q p = (q)defined on an open subset W of Rn and satisfying the following topological-differential properties:

necessary and sufficient for a subset A of E to inherit from the affine environment astructure of affine space modelled on a vector subspace A of E. Familiar examples are thestraight lines and the planes (i.e. the affine subspaces of dimension 1 and 2, respectively).

4 A repeated index, in lower and upper position, will denote summation.5 Here we refer to the subspace topology of A .6 Recall that a mapping like is C- differentiable, iff it is continuous and admits

continuous partial derivatives of any order. Its differential at q is the linear map

dq : Rn E : v = (vh) v = vh p

qh

q

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44 5 Appendix

1. : W U is a homeomorphism;2. : W E is C- differentiable, with injective differential dq at each

q W .

Remark that dq is injective, iff its image

Im dq = Span

p

qh

q

(spanned by the first-order partial derivatives of ) is an n-dimensional vector

subspace of E.Hence n m .

Atlas of charts

Owing to the definition, a manifold Q E is covered by an atlas of opensubsets U described in E by means of local parametrizations . Each pointp U is then given a unique n-tuple of coordinates q = 1(p) W by theinverse map 1 : U W , which is called an n-dimensional local chart (orlocal coordinate system) on Q (it is said to be global in the special case of itscoordinate domain

Ubeing equal to the whole Q ). 7

Locally Euclidean topology

From a topological point of view, Q is a locally Euclidean subspace of E, inthe sense that it is covered by open subsets (namely, the coordinate domains)which, owing to property 1, are all homeomorphic to open subsets of Euclideanspace Rn .

Smoothness

From a differential point of view, Q is a smooth subspace of E, in the sensethat it admits, owing to property 2, an n-dimensional tangent vector spaceat each one of its points, as will now be shown.

Smooth curves and tangent vectors

Recall that tangency is a local (infinitesimal) concept, linked to derivationas follows.

7 In the sequel, too will be called a chart.

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5 Appendix 45

Consider a smooth curve of E, i.e. a C

- differentiable E-valued map : I E: t (t) = p(t)

defined on an open interval I R ( Im will be called the orbit of ).The vector p(t) E obtained from by derivation at any t I, is said

to be the free vector tangent at p(t) to . 8

Tangent vector spaces

Now consider the smooth curves of

Ewith orbits lying on Q, i.e.

Im Q

(also referred to as smooth curves of Q and denoted by : I Q ), andpassing through a given point p Q , i.e.

p Im

A vector of E will be said to be a free vector tangent at p to Q , if it istangent at p to one of the above curves.

The set

TpQ Eof all the free vectors tangent at p to Q will be called tangent vector space ofQ at p , by virtue of the following result.

Proposition 19 TpQ is an n-dimensional vector subspace of E.

Proof Follows from property 2, owing to the followingLemma For any chart : W U at p (i.e. U p = (q) , with q W),

TpQ = Im dq ()

(i) Let v TpQ , i.e.v = p(to)

for some smooth curve : t I p(t) Q through p(to) = p (with to I).8 Think of the graphic representation of derivative p(t) := limt0

1t

p(t + t) p(t)

when it does not vanish as an oriented segment, limit of a secant segment, attached atp(t) .

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46 5 Appendix

As U is an open neighbourhood of p(to) = p in Q , there exists (by continuity)an open neighbourhood Io of to in I s.t. (Io) U. As a consequence, cabe given the local C coordinate expression

= 1 |Io : Io W : t q(t) = 1(p(t))By derivation at to of p(t) = (q(t)) , we obtain

v = p(to)

= qh(to)p

qh

q(to)

= dq(to)(q(to))

= dq(v) , v := q(to) Rn

Hence v Im dq.(ii) Now let v Im dq, i.e.

v = dq(v) , v RnChoose to R and consider the affine map

: R Rn : t q(t) := q+ (t to)vAs W is an open neighbourhood of q(to) = q in Rn , there exists (by continu-ity) an open neighbourhood Io of to in R s.t. (Io)

W . As a consequence,

= |Io : Io U: t p(t) = (q(t))is a smooth curve of Q through

p(to) = (q(to)) = (q) = p

whose derivative at to is

p(to) = qh(to)

p

qh

q(to)

= dq(to)(q(to))

= dq(v)

= vHence v TpQ .

Remark thatv TpQ v = (dq)1(v) Rn

is the linear isomorphism which takes any vector v TpQ to its componentsv Rn in the basis

pqh

q

of TpQ (called linear components of v in ).

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5 Appendix 47

Open submanifolds

Trivial examples of submanifolds of E are the open subets.

Any system of affine coordinates on E restricted to an open subset U Edetermines an m-dimensional global chart on U, which is therefore an m-dimensional embedded submanifold of E and, for all p U,

TpU = E

For instance, if

g = (g1, . . . , g) : E R

: p g(p) = (g1(p), . . . , g(p))is a continuous map, then

U := g1(R+) = {p E | g(p) (R+)}= {p E | g1(p) > 0, . . . , g(p) > 0}

(inverse image by g of the open subset (R+) R ) is an open submanifoldof E.

Implicit function theorem

Non-trivial examples of submanifolds of E arise from the well known ImplicitFunction Theorem, concerning loci described by means of algebraic equations.

Proposition 20 Let

f = (f1, . . . , f ) : U E R : p f(p) = (f1(p), . . . , f (p))be a differentiable map, defined on an open subset U E (say U = g1(R+) )and taking values in R ( < m := dim E), with surjective differential dpf :E R at each point p of

Q := f1(0) = {p U | f(p) = 0} = {p U | f1(p) = 0, . . . , f (p) = 0}= {p E | f1(p) = 0, . . . , f (p ) = 0, g1(p) > 0, . . . , g(p) > 0}

Q is then an n-dimensional embedded submanifold of E withn := m

and, for all p Q ,TpQ = Ker dpf

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48 5 Appendix

Proof (i) The above mentioned theorem of Analysis states that, for any p Q ,the fact of dpf being surjective implies the existence of an open neighbourhoodU of p in Q , an open subset W Rn and a -tuple of real-valued functions1 : W R, . . . , : W R s.t.

U= Im

where

: W Eis a map defined (with a suitable choice of affine coordinates : Rm E) by

(x1, . . . , xn) ( x1, . . . , xn, 1(x1, . . . , xn), . . . , (x1, . . . , xn) )

and satisfying properties 1 and 2, i.e. an n-dimensional chart on Q . 9

(ii) As is composable with f and f = 0 , we have

dpf dq = dq(f ) = 0

(with p = (q) ) and then

Im dq Ker dpfMoreover, the surjectivity of dpf and the injectivity of dq imply the dimen-sional result

dimKer dpf = dim E dimIm dpf = m = n = dim Im dq

So we obtain

Im dq = Ker dpf

which, owing to (), completes our proof.

9 So, on 1(U) , the algebraic equations

f1

(x1, . . . , xn, xn+1, . . . , xn+)

= 0, . . . , f

(x1, . . . , xn, xn+1, . . . , xn+)

= 0

implicitly define xn+1, . . . , xn+ as functions of (x1, . . . , xn) W , i.e.

xn+1 = 1(x1, . . . , xn), . . . , xn+ = (x1, . . . , xn)

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5 Appendix 49

5.2 Tangent bundle and differential equationsThe tangent bundle of a manifold is the first geometric arena of differentialequations.

Tangent bundle and canonical projection

The tangent bundleof an n-dimensional smooth manifold Q E is the disjointunion of its tangent vector spaces, i.e.

T Q =

pQ

{p} TpQ

(for any p Q , the set {p} TpQ is made up of all the attached vectorstangent at p to Q ).

The canonical projection of T Q onto its base manifold Q is the map

Q : T Q Q : (p, v) p

Vector fibre bundle structure

T Q is a vector fibre bundle over Q , in the sense that its fibre {p} TpQ overeach p

Q is a vector space (canonically isomorphic to TpQ ).

Such fibres are glued together by natural charts (arising from the charts : W U of Q )

1 : (q, v) W Rn (p, v) = ((q), dq(v)) 1Q (U)which give T Q the structure of a 2n-dimensional smooth manifold embeddedin TE= E E. 10

Tangent lift

The tangent lift of a smooth curve

: I Q : t (t) = p(t)living in Q E, is the smooth curve

: I T Q : t (t) = (p(t), p(t))living in T Q TE.

10 Remark that, for any open subset U E (and then for U = E as well), T U = U E.Recall that TE= E E is a Euclidean affine space, modelled on E E.

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Differential equations in implicit form

From the very definition of tangent vector spaces, it follows that T Q is theregion of TE swept by the tangent lifts of all the smooth curves of Q (i.e.covered by the orbits of such lifts).

As a consequence, if a subset

D T Q

is assigned (by prescribing some geometric or algebraic property), the problemmay arise of determining the smooth curves of Q whose tangent lifts live

in D , i.e.Im D ()

that is to say,

(t) = (p(t), p(t)) D , t I ()Such a D will be called (autonomous) first-order differential equation in

implicit form on Q . 11

Integral curves

Any solution to the above problem, i.e. any smooth curve of Q satisfyingcondition (), is called an integral curve of D (or maximal integral curve, ifit is not restriction of any other integral curve).

Integrable part

The problem of establishing the existence of integral curves is the same as thatof determining the integrable part of D , which is the region

D(i)

Dswept by the tangent lifts of all its integral curves.

D is said to be integrable, if

= D = D(i)

11 We shall not consider the case of a non-autonomous equation D J1Q , subset ofthe jet bundle J1Q := R T Q , whose solutions are the smooth curves of Q with jetextension j1 : t I j1(t) := (t, p(t), p(t)) J1Q living in D , i.e. Im j1 D .

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Vector fields and normal form

The main case of integrability is that of a differential equation D reducible tonormal form, that is, one for which there exists a smooth vector field

X : Q T Q : p (p, X(p))

on Q 12 s.t.

D = Im X = {(p, v) T Q | v = X(p)}For such an equation, condition () reads

= X

that is to say,

p(t) = X(p(t)) , t I

Cauchy problems and determinism

The integrability of

D = Im X =

is a consequence of the following fundamental property of normal equations(here stated without proof).

Determinism theorem: For any to R and po Q , there exists a uniquemaximal solution to Cauchy problem (D, to , po) , i.e. a unique maximal inte-gral curve : t I p(t) Q of D , with I to , satisfying initial conditionp(to) = po (all of the other solutions to the problem being restrictions of themaximal one).

Indeed, for any (po, vo) D , i.e. po Q and vo = X(po) , we obtain

(po, vo) = (p(to), X(p(to))) = (p(to), p(to)) Im o D(i)

whence

D = D(i)

12 X is said to be smooth, if in any chart : W U it admits C linear components

q W (dq)1

X((q)) Rn

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5.3 Second tangent bundle and second-orderdifferential equations

The second tangent bundle of a manifold is the geometric arena of second-orderdifferential equations.

Second tangent lift and second tangent bundle

Consider the tangent bundle

T T Q T TE

of T Q TE.Recall that T T Q is the region of T TE= (E E) (E E) swept by the

tangent lifts of all the smooth curves of T Q , the tangent lift of any such curve

c : I T Q : t c(t) = (p(t), v(t))

being given by

c : I T T Q : t c(t) = (p(t), v(t); p(t), v(t))

In particular, the second tangent lift of a smooth curve

: I Q : t (t) = p(t)

living in Q , is the tangent lift of

: I T Q : t (t) = (p(t), p(t))

i.e. the smooth curve 13

: I T2Q : t (t) = (p(t), p(t); p(t), p(t))

living in the region T2Q T T Q , called second tangent bundle of Q, definedby

T2Q = {(p, v; u, w) T T Q | u = v}13 p(t) will denote the derivative of p(t) .

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5 Appendix 53

Affine fibre bundle structure

Remark thatT2Q =

(p,v)TQ

{(p, v)} {v} T2(p,v)Q

withT2(p,v)Q := {w E | (p, v; v, w) T T Q}

is an affine fibre bundle over T Q , in the sense that its fibre {(p, v)} {v} T2(p,v)Q

over each (p, v) T Q is an affine space (canonically iso-

morphic to T2(p,v)Q ), as will now be shown.

Proposition 21 For any (p, v) T Q , T2(p,v)Q is an n-dimensional affinesubspace of E modelled on TpQ , i.e.

T2(p,v)Q = z + TpQ

for some z E.

Proof Follows from property (), owing to the followingLemma For any natural chart 1 : WRn 1Q (U) at (p, v) (i.e. 1Q (U) (p, v) = ((q), dq(v)) with (q, v)

W

Rn ),

T2(p,v)Q = z(q, v) + Im dq ()

where z(q, v) := vhvk 2p

qhqk

q

.

(i) Let w T2(p,v)Q , that is,(p, v; v, w) T T Q

or(p, v; v, w) = c(to) = (p(to), v(to); p(to), v(to))

for some smooth curve c : t I c(t) = (p(t), v(t)) T Q and to I.As Uis an open neighbourhood of p = p(to) in Q , there exists (by continuity)an open neighbourhood Io of to in I s.t. p(t) U for all t Io . Then c|Iocan be given the C coordinate expression

p(t) = (q(t)) , v(t) = dq(t)(v(t)) = vh(t)

p

qh

q(t)

withq(to) = q , q(to) = v(to) = v

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By derivation at to , we obtainw = v(to)

= vh(to)qk(to)

2p

qhqk

q(to)

+ vh(to)p

qh

q(to)

= vhvk2p

qhqk

q

+ whp

qh

q

, w := v(to)

= z(q, v) + dq(w)

Hence w z(q, v) + Im dq.(ii) Now let w

z(q, v) + Im dq, that is,

w=z(q, v) + dq(w)

for some w Rn .Choose to R and consider the map

: R Rn : t q(t) := q+ v(t to) + 12

w(t to)2

satisfyingq(to) = q , q(to) = v , q(to) = w

As W is an open neighbourhood of q(to) = q in Rn , there exists (by continu-ity) an open neighbourhood Io of to in R s.t. (Io)

W . As a consequence,

= |Io : Io U: t p(t) = (q(t))is a smooth curve of Q through

p(to) = (q(to)) = (q) = p

whose derivatives at to are

p(to) = qh(to)

p

qh

q(to)

= dq(to)(q(to))

= dq(v)

= v

and

p(to) = qh(to)q

k(to)2p

qhqk

q(to)

+ qh(to)p

qh

q(to)

= vhvk2p

qhqk

q

+ whp

qh

q

= z(q, v) + dq(w)

= w

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5 Appendix 55

So

(p, v; v, w) = (p(to), p(to); p(to), p(to)) = (to) Im T2Q T T Q

Hence w T2(p,v)Q .

Remark that

w T2

(p,v)Q w = (dq)1

(w z(q, v)) Rn

is the affine isomorphism which takes any vector w T2(p,v)Q to its affinecoordinates w Rn in the frame

z(q, v),

pqh

q

of T2(p,v)Q (called affine

components of w in ).

Second-order differential equations in implicit form

Part (ii) of the above proof shows as well that T2Q is the region of T T Qswept by the second tangent lifts of all the smooth curves of Q .

As a consequence, if a subset

D T2Q

is assigned (by prescribing some geometric or algebraic property), the problemmay arise of determining the smooth curves : t I p(t) Q whosesecond tangent lifts live in D , i.e.

Im D ()that is to say,

(t) = (p(t), p(t); p(t), p(t)) D , t I ()

Such a D will be called (autonomous) second-order differential equationin implicit form on Q .

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Integral curves and base integral curves

A second-order differential equation on Q , say D T2Q T T Q , is a first-order differential equation on T Q as well, whose integral curves exhibit thefollowing peculiar property.

Proposition 22 Each integral curve c of D T2Q is the tangent lift of itsown projection Q c , i.e.

(Q c) = c

Proof Letc : t

I

c(t) = (p(t), v(t))

T Q

be a smooth curve of T Q , and

Q c : t I Q(c(t)) = p(t) Qits projection onto Q .If c is an integral curve of D , it satisfies, for all t I,

c(t) = (p(t), v(t); p(t), v(t)) Dwhence

p(t) = v(t)

that is,(Q c)(t) = (p(t), p(t)) = (p(t), v(t)) = c(t)

which is our claim.

The base integral curves = Qc of D projections of the integral curvesc of D are the solution curves of problem (), as will now be shown.Proposition 23 A smooth curve of Q is a base integral curve of D , iff itis a solution curve of problem ().Proof (i) Let Im D . Putting c := , whence Q c = Q and c = ,we obtain Q

c = and Im c

D.

(ii) Conversely, let = Q c and Im c D . From Prop. 22, w

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