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Introduction to the

Geometry of Classical Dynamics

Renato Grassini

Dipartimento di Matematica e Applicazioni

Universita di Napoli Federico II

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

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

Keywords: smooth manifolds, implicit differential equations, d’Alembert’sprinciple, Lagrangian and Hamiltonian dynamics

Published by Hikari Ltd

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-d’Alembertian 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 equationjust 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-d’Alembertian 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).

Contents

Preface iii

1 From Newton to d’Alembert 11.1 The data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

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

1.2 The question . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Smooth motions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Dynamically possible motions . . . . . . . . . . . . . . . . . . . . 3

1.3 The answer after Newton . . . . . . . . . . . . . . . . . . . . . . 4Newton’s law of constrained dynamics . . . . . . . . . . . . . . . . 4

1.4 d’Alembert’s reformulation . . . . . . . . . . . . . . . . . . . . . 4d’Alembert’s principle of virtual works . . . . . . . . . . . . . . . . 4

1.5 d’Alembert’s implicit equation . . . . . . . . . . . . . . . . . . . 5Tangent dynamically possible motions . . . . . . . . . . . . . . . . 5d’Alembert equation . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 From d’Alembert to Lagrange 82.1 Integrable part of d’Alembert equation . . . . . . . . . . . . . . 8

Restriction of d’Alembert 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

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 333.1 Legendre transformation . . . . . . . . . . . . . . . . . . . . . . 33

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

4 Concluding remarks 394.1 Inertia and force . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . 404.3 Geometrizing physical fields . . . . . . . . . . . . . . . . . . . . 414.4 Geometrical dynamics . . . . . . . . . . . . . . . . . . . . . . . 41

5 Appendix 425.1 Submanifolds of a Euclidean space . . . . . . . . . . . . . . . . 42

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

vi Table of Contents

Implicit function theorem . . . . . . . . . . . . . . . . . . . . . . 475.2 Tangent bundle and differential equations . . . . . . . . . . . . . 49

Tangent 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 . 52Second 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

Chapter 1

From Newton to d’Alembert

In this chapter, we shall recall the problem of classical particle dynamics,Newton’s answer to the problem and d’Alembert’s 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 – inthe 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 ν particles, a position – or con-figuration – in E3 is defined by a single point of E := E3

ν (3ν-dimensional,Euclidean, affine space, modelled on E := E3

ν ). 1

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

v ·w :=ν∑

i=1

vi ·wi

1

2 1 From Newton to d’Alembert

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 ⊂ E

called configuration space of the system in E3 .Q generally consists of all the points p ∈ E satisfying a number κ < 3ν

of independent scalar equalities fα(p) = 0 (α = 1, . . . , κ), called two-sidedconstraints, and/or some strict scalar inequalities gβ(p) > 0 (β = 1, . . . , µ),called strict one-sided constraints. Under usual hypotheses of regularity onthe constraints, Q is an embedded submanifold of E , whose dimension n :=dim Q = 3ν − κ – 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 by

m := (m1, . . . ,mν)

Force field

The ‘force’ – δυναµις – 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 ⊂ TE → E : (p,v) 7−→ F(p,v) = (F1(p,v), . . . ,Fν(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 7→ 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.

1 From Newton to d’Alembert 3

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 dynamicscan 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

γ : I ⊂ R → E : t 7→ γ(t) = p(t)

Such a curve γ 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), . . . , pν(t))of the particles, their velocities p(t) = (p1(t), . . . , pν(t)) and their accelera-tions p(t) = (p1(t), . . . , pν(t)) (as well as the derivatives of any order) varycontinuously.

Dynamically possible motions

Smooth dynamics basically deals with the ‘time-evolution problem’ – in theunknown γ – 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 δυναµις ? ”

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 ).


1.3 The answer after Newton

After Newton, the answer to the above ‘predictive’ question is given by thefollowing law.

Newton’s law of constrained dynamics

A smooth curveγ : I → E : t 7→ γ(t) = p(t)

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

p(t) ∈ Q

m p(t) = F(p(t), p(t)) + Φ(t)

Φ(t) ∈ T⊥p(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 Newton’s law with a right hand sideencompassing the possible ‘dynamical effects’ of the contraints, expressed byan ‘unknown’ constraint reaction Φ(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 d’Alembert’s reformulation

After d’Alembert, the unknown constraint reaction can be ‘cancelled’ fromNewton’s law of constrained dynamics as follows.

d’Alembert’s 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) ∈ T⊥p(t)Q

4 For any µ = (µ1, . . . , µν) ∈ Rν and w = (w1, . . . ,wν) ∈ E , we shall put µw :=(µ1w1, . . . , µνwn) . Moreover, for any p ∈ Q , T⊥p 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 .


that is, (F(p(t), p(t))−m p(t)

)· δp = 0 , ∀ δp ∈ Tp(t)Q

called d’Alembert’s principle of virtual works (since the inner product thereindefines the work of active force F(p(t), p(t)) and inertial force −mp(t) alongany virtual displacement δp , i.e. any ‘infinitesimal’ displacement tangent toQ and therefore virtually allowed by the constraints). 6

So a smooth curve

γ : I → E : t 7→ γ(t) = p(t)

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

p(t) ∈ Q , m p(t) · δp = F(p(t), p(t)) · δp , ∀ δp ∈ Tp(t)Q (�)′

1.5 d’Alembert’s implicit equation

From the mathematical point of view, condition (�)′ shows that determiningthe DPM s 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

Ifγ : I → E : t 7→ γ(t) = p(t)

is DPM of S , its tangent lift

γ : I → TE : t 7−→ γ(t) = (p(t), p(t))

will be called a tangent dynamically possible motion (TDPM ) of S .

6 Owing to footnotes 1 and 4, d’Alembert’s principle reads

ν∑i=1

(Fi(p(t), p(t))−mipi(t)

)· δpi = 0 , ∀ δp = (δp1, . . . , δpν) ∈ Tp(t)Q

Remark that, if Q is an open submanifold of E (absence of two-sided constraints), onehas T⊥p Q = E⊥ = {0} for all p ∈ Q (absence of constraint reaction) and then d’Alembert’sprinciple simply reads

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


DPM s and TDPM s bijectively correspond to one another, since the tangentlift operator γ 7→ γ is obviously inverted by the base projection operatorγ 7→ γ . Trough such a bijection, the problem of determining the DPM sproves to be naturally equivalent to that of determining the TDPM s.

Owing to (�)′, a smooth curve

c : I → TE : t 7−→ 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) · δp = F(p(t),v(t)) · δp , ∀ δp ∈ Tp(t)Q (�)

d’Alembert 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 d’Alembert equa-tion

Dd′Al := {(p,v;u,w) ∈ TTE | p ∈ Q , u = v , F(p,v)−mw ∈ T⊥p Q }

= {(p,v;u,w) ∈ TTE | p ∈ Q , u = v , mw · δp = F(p,v) · δp , ∀ δp ∈ TpQ }⊂ T 2E

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

Proof Recall that a smooth curve

c : I → TE : t 7−→ c(t) = (p(t),v(t))

is an integral curve of Dd′Al , iff its tangent lift

c : I → TTE : t 7−→ c(t) = (p(t),v(t); p(t), v(t))

satisfies conditionIm c ⊂ Dd′Al

that is, for all t ∈ I ,

c(t) = (p(t),v(t); p(t), v(t)) ∈ Dd′Al

which is exactly condition (�), characterizing the TDPM s.

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


Also recall that a smooth curve

γ : I → TE : t 7→ γ(t) = p(t)

is a base integral curve of Dd′Al , iff its second tangent lift

γ : I → TTE : t 7−→ γ(t) = (p(t), p(t); p(t), p(t))

satisfies conditionIm γ ⊂ Dd′Al

that is, for all t ∈ I ,

γ(t) = (p(t), p(t); p(t), p(t)) ∈ Dd′Al

which is exactly condition (�)′, characterizing the DPM s.

�

Chapter 2

From d’Alembert to Lagrange

Dynamics is now a problem of integration, i.e. determination and/or qualita-tive analysis of the integral curves of d’Alembert 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 TQ ), which will naturally be obtained and thoroughly discussed.

2.1 Integrable part of d’Alembert equation

As to the integration of Dd′Al , the first step is to extract its integrable part,i.e. the region

D(i)d′Al ⊂ Dd′Al

swept 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)d′Al , it is quite natural to start from

the following remark.

Owing to Prop.1, the base integral curves (if any) of Dd′Al are constrainedto live in Q (see condition (�)′) and then their tangent lifts, i.e. the integralcurves, live in TQ . As a consequence, the tangent lifts of the integral curveslive in TTQ , that is to say,

D(i)d′Al ⊂ TTQ

So we obtain

D(i)d′Al ⊂ Dd′Al ∩ TTQ

8

2 From d’Alembert to Lagrange 9

Restriction of d’Alembert equation

The above result suggests focusing on the ‘restriction’ of Dd′Al obtained viaintersection with TTQ , i.e. on the first-order differential equation in implicitform on TQ (second-order on Q )

DLagr := Dd′Al ∩ TTQ

which will be called Lagrange equation.Note that Lagrange equation is an effective restriction of d’Alembert equa-

tion (i.e. DLagr & Dd′Al ) in presence, and only in presence, of two-sidedconstraints (when dim Q < dim E ), as is shown by the following proposition.

Proposition 2 DLagr & Dd′Al , 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 & Dd′Al . Indeed, we can choosep ∈ Q , v ∈ E − TpQ (whence (p,v) 6∈ TQ ), u = v and w = 1

m(F(p,v) +

Φ) with Φ ∈ T⊥p Q (whence F(p,v) − mw ∈ T⊥

p Q ) and clearly we obtain(p,v;u,w) ∈ Dd′Al , (p,v;u,w) 6∈ TTQ and then (p,v;u,w) 6∈ DLagr .Conversely, let dim Q = dim E (whence TpQ = T 2

(p,v)Q = E for all (p,v) ∈TQ ). We shall prove that, under the above hypothesis, Dd′Al ⊂ DLagr .Indeed, for any (p,v;u,w) ∈ Dd′Al , we have p ∈ Q , v ∈ E = TpQ ,u = v and w ∈ E = T 2

(p,v)Q , that is, (p,v;u,w) ∈ T 2Q ⊂ TTQ and

then (p,v;u,w) ∈ DLagr .�

Extraction of the integrable part

By extracting DLagr from Dd′Al , via intersection of the latter with TTQ , we

just obtain D(i)d′Al 6= ∅ , as is shown by the following proposition.

Proposition 3 D(i)d′Al = DLagr 6= ∅

Proof First we remark that Dd′Al and DLagr are equivalent equations – i.e.

they have the same integral curves – since, owing to inclusions D(i)d′Al ⊂ DLagr ⊂

Dd′Al , condition Im c ⊂ Dd′Al (i.e. Im c ⊂ D(i)d′Al ) implies Im c ⊂ DLagr and,

conversely, Im c ⊂ DLagr implies Im c ⊂ Dd′Al .

That amounts to saying D(i)d′Al = D(i)

Lagr .

Then we anticipate that DLagr is integrable, i.e. ∅ 6= DLagr = D(i)Lagr .

Hence our claim.�

10 2 From d’Alembert 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 traditionallocal 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) ∈ TTQ | u = v , mw·δp = F(p,v)·δp , ∀ δp ∈ TpQ } ⊂ T 2Q

From 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] : T 2Q → T ∗Q : (p,v;v,w) 7−→ (p, [m](p,v,w))

on T 2Q , called covector inertial field, whose value

[m](p,v,w) := (mw) · |TpQ ∈ T ∗pQ

at any (p,v;v,w) ∈ T 2Q is the virtual work of mw . 1

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

F : TQ → T ∗Q : (p,v) 7−→ (p, F (p,v))

on TQ , called covector force field, whose value

F (p,v) := F(p,v) · |TpQ ∈ T ∗pQ

at any (p,v) ∈ TQ is the virtual work of F(p,v) . 2

1 For any u ∈ E , we define u · ∈ E∗ by putting u · : v ∈ E 7→ u · v ∈ R . Then, for anyp ∈ Q , the restriction of u · to TpQ yields u · |TpQ ∈ T ∗p Q .

2 The virtual work of a positional force field f can be regarded as an ordinary 1-form onQ , namely f : p ∈ Q 7→ (p, f(p)) ∈ T ∗Q , f(p) := f(p) · |TpQ ∈ T ∗p Q .


Proposition 4 Lagrange equation can be given the covector formulation

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

Proof Just notice that condition

mw · δp = F(p,v) · δp , ∀ δp ∈ TpQ

means

(mw) · |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) ∈ TQ , the algebraic equation [m](p,v,w) = F (p,v) should beuniquely solvable with respect to the unknown w ∈ T 2

(p,v)Q . As the latteronly appears in the left hand side of the equation, the above property is to bechecked through a thorough investigation of [m] .

The following geometric considerations – showing that such a semi-basic1-form on T 2Q is the transformed of a suitable semi-basic 1-form on TQthrough a distinguished semi-spray – will prove to be crucial.

Remark that [m] is the semi-basic 1-form ‘induced’ on T 2Q by the Eu-clidean metric

gm : E → E∗ : u 7−→ gm(u) := (mu) ·

(positive definite, symmetric, linear map), since its value at any (p,v;v,w) ∈T 2Q is

[m](p,v,w) = gm(w)∣∣∣TpQ

∈ T ∗pQ

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

g : TQ → T ∗Q : (p,v) 7−→ (p, gp(v))

induced on TQ by gm , whose value at any (p,v) ∈ TQ is

gp(v) := gm(v)∣∣∣TpQ

∈ T ∗pQ


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

K : TQ → R : (p,v) 7→ K(p,v) :=1

2〈gp(v) | v〉 =

1

2mv · v

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

ΓK : TQ → T 2Q : (p,v) 7→ (p,v;v, ΓK(p,v))

called Riemannian spray, uniquely determined by the following property.

Proposition 5 There exists one, and only one, semi-spray ΓK on TQ s.t.,for all (p,v) ∈ TQ ,

gm(ΓK(p,v))∣∣∣TpQ

= 0

Proof (i) Unicity First remark that, for any (p,v) ∈ TQ , the map

w ∈ T 2(p,v)Q

(α)7−→ gm(w)∣∣∣TpQ

∈ T ∗pQ

is injective, since

gm(w1)∣∣∣TpQ

= gm(w2)∣∣∣TpQ

–with w1,w2 ∈ T 2(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 → T ∗pQ is a linear isomorphism,

w1 = w2

So, if there exists a vector in T 2(p,v)Q whose image through (α) is zero, it is

unique.

(ii) Existence For any (p,v) ∈ TQ , put

ΓK(p,v) := w + u ∈ T 2(p,v)Q


withw ∈ T 2

(p,v)Q , u := − g−1p

(gm(w)

∣∣∣TpQ

)∈ TpQ

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

gm(ΓK(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

�

ΓK transforms g (semi-basic 1-form on TQ ) into

[K] : T 2Q → T ∗Q : (p,v;v,w) 7→ (p, [K](p,v,w))

(semi-basic 1-form on T 2Q ) by putting, for any (p,v;v,w) ∈ T 2Q ,

[K](p,v,w) := gp(w − ΓK(p,v)) ∈ T ∗pQ

[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) ∈ TTQ | 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) ∈ T 2Q , from Prop. 5 we obtain

[K](p,v,w) := gp(w − ΓK(p,v))

= gm(w − ΓK(p,v))∣∣∣TpQ

= gm(w)∣∣∣TpQ

− gm(ΓK(p,v))∣∣∣TpQ

= gm(w)∣∣∣TpQ

= [m](p,v,w)

�


Normal form

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

Consider the vertical force field

∆F := g−1 ◦ F : TQ → TQ : (p,v) 7→ (p, ∆F (p,v))

∆F (p,v) := g−1p (F (p,v)) ∈ TpQ

and the semi-spray

Γ := ΓK + ∆F : TQ → T 2Q : (p,v) 7→ (p,v;v, Γ(p,v))

Γ(p,v) := ΓK(p,v) + ∆F (p,v) ∈ T 2(p,v)Q

Proposition 7 Lagrange equation can be put in the normal form

DLagr = Im Γ = {(p,v;u,w) ∈ TTQ | u = v , w = Γ(p,v)}

Proof Just notice that covector equality

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

readsgp(w − ΓK(p,v)) = F (p,v)

w − ΓK(p,v) = g−1p (F (p,v))

w = ΓK(p,v)) + ∆F (p,v)

w = Γ(p,v)

�

Integral curves

The condition characterizing the integral curves of Lagrange equation can nowbe formulated as follows.

Letc : I → TQ : t 7→ c(t) = (p(t),v(t))

be a smooth curve of TQ and

τQ ◦ c : I → Q : t 7→ (τQ ◦ c)(t) = p(t)

its projection onto Q . 3

3 The tangent lift of τQ ◦ c will be denoted by (τQ ◦ c)� .


Proposition 8 c is an integral curve of DLagr , iff

(τQ ◦ c)� = c , [K] ◦ c = F ◦ c (◦)

or, in normal form,c = Γ ◦ c (•)

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

Im 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 (∗) reads

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

that is,

(τQ ◦ 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) = Γ(p(t),v(t))

that is,

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

= (p(t),v(t);v(t), Γ(p(t),v(t)))

= (Γ ◦ c)(t)

That proves our second claim.�


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

Letγ : I → Q : t 7→ γ(t) = p(t)

be a smooth curve of Q and

[K] ◦ γ : I → T ∗Q

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

∇γ

dt:= g−1 ◦ [K] ◦ γ : I → TQ

Proposition 9 γ is a base integral curve of DLagr , iff

[K] ◦ γ = F ◦ γ (◦)′

or, in vector formulation,∇γ

dt= ∆F ◦ γ

or, in normal form,γ = Γ ◦ γ (•)′

Proof Recall that a base integral curve of Lagrange equation is the projectionγ = τQ ◦ c of an integral curve c (smooth curve of TQ satisfying condition(◦) or the equivalent (•)).

(i) Now, if γ is a base integral curve, condition (◦) implies γ = (τQ ◦ c)� = c–whence γ = c – and then (◦)′. Conversely, if γ satisfies condition (◦)′, then itis obviously a base integral curve (projection of c := γ satisfying (◦)). Clearly,condition (◦)′ is equivalent to

g−1 ◦ [K] ◦ γ = g−1 ◦ F ◦ γ

that is,∇γ

dt= ∆F ◦ γ

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) .


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

∇γ

dt: t ∈ I

γ7−→ (p(t), p(t); p(t), p(t)) ∈ T 2Q

[K]7−→(p(t), gp(t) (p(t)− ΓK(p(t), p(t)))

)∈ T ∗Q

g−1

7−→(p(t), p(t)− ΓK(p(t), p(t))

)∈ TQ

and

γ − ΓK ◦ γ : t ∈ I 7→ (p(t), p(t)− ΓK(p(t).p(t))) ∈ TQ

that is,∇γ

dt= γ − ΓK ◦ γ

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

γ = ΓK ◦ γ + ∆F ◦ γ

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] ◦ γ = 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 6= 0 is then that of deviating theDPM s of S from the inertial trend, by giving them a non-vanishing Rieman-nian geodesic curvature, namely [K] ◦ γ = F ◦ γ .

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.


Preliminaries

Consider the coordinate domain U = Im ξ of a chart ξ : q ∈ W 7→ ξ(q) ∈ U ,expressing the points p ∈ U ⊂ Q in function of coordinates q ∈ W ⊂ Rn

(with n := dim Q ). 5

To any smooth curve γ : t ∈ I 7→ γ(t) = p(t) ∈ Q living in U , i.e.satisfying

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

q = q(t) ∈ W

related to γ by p(t) = ξ(q(t)) , also denoted

p = ξ(q) (1)

(dependence from time t is understood).

To the first tangent lift c = γ , that is,

p = p(t) ∈ U , v = v(t) ∈ Tp(t)Q

withv = p

there corresponds in ξ a smooth coordinate expression

q = q(t) ∈ W , v = v(t) ∈ Rn

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

v = v(q, v) := vh ∂p

∂qh

∣∣∣q

(2)

wherev = q

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

∂v

∂vh

∣∣∣(q,v)

=∂p

∂qh

∣∣∣q,

∂v

∂qh

∣∣∣(q,v)

= vk ∂2p

∂qh∂qk

∣∣∣q

=d

dt

∂p

∂qh

∣∣∣q

(3)

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

∂p∂q1

∣∣∣q, . . . , ∂p

∂qn

∣∣∣q

)provide a basis of TpQ .

6 A repeated index, in upper and lower position, denotes summation over (1, . . . , n).


To the second tangent lift c = γ , that is,

p = p(t) ∈ U , v = v(t) ∈ Tp(t)Q , w = w(t) ∈ T 2(p(t),v(t))Q

with

v = p , w = v = p

there corresponds in ξ a smooth coordinate expression

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

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

w = w(q, v, w) := wh ∂p

∂qh

∣∣∣q+ vhvk ∂2p

∂qh∂qk

∣∣∣q

where

w = v = q

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

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

Fh(q, v) := (F ◦ γ)h = (F (p,v))h =

⟨F (p,v) | ∂p

∂qh

∣∣∣q

⟩= F(p,v) · ∂p

∂qh

∣∣∣q

and the components of ∆F ◦ γ = g−1 ◦ F ◦ γ will be denoted by

F i(q, v) := (∆F ◦ γ)i = (∆F (p,v))i = (g−1p (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

⟩= m

∂p

∂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.


Proposition 10 A smooth curve γ : t ∈ I 7→ p = ξ(q(t)) ∈ U ⊂ Q – livingin the coordinate domain U of a chart ξ 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)

− ∂K

∂qh

∣∣∣(q,v)

= Fh(q, v) (◦)h

or, in normal form, 7

q = v , vi = −{

i

jk

}q

vjvk + F i(q, v) (•)i

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

([K] ◦ γ)h = (F ◦ γ)h (◦)′h

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

([K] ◦ γ)h := ([K](p,v,w))h

=

⟨[K](p,v,w) | ∂p

∂qh

∣∣∣q

⟩= mw(q, v, w) · ∂p

∂qh

∣∣∣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

}:=

12gih(∂jgkh + ∂kghj − ∂hgjk)

with∂ighk :=

∂ghk

∂qi


=d

dt

∂

∂vh

∣∣∣(q,v)

(1

2mv · v

)− ∂

∂qh

∣∣∣(q,v)

(1

2mv · v

)=

d

dt

∂K

∂vh

∣∣∣(q,v)

− ∂K

∂qh

∣∣∣(q,v)

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

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

γi = (Γ ◦ γ)i (•)′ i

where

γi = wi

= vi

and

(Γ ◦ γ)i = (ΓK ◦ γ + ∆F ◦ γ)i

= (ΓK ◦ γ)i + (∆F ◦ γ)i

= ΓiK(q, v) + F i(q, v)

The components F i(q, v) := (∆F ◦ γ)i have already been shown in the abovepreliminaries. The components Γi

K(q, v) := (ΓK ◦ γ)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) vhvk

whence

∂K

∂qh

∣∣∣(q,v)

=1

2(∂hgjk)q vjvk

∂K

∂vh

∣∣∣(q,v)

= ghk(q)vk

and

∂2K

∂vh∂qk

∣∣∣(q,v)

= (∂kghj)q vj

∂2K

∂vh∂vk

∣∣∣(q,v)

= ghk(q)


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

([K] ◦ γ)h = ([K](p,v,w))h

=∂2K

∂vh∂vk

∣∣∣(q,v)

wk +∂2K

∂vh∂qk

∣∣∣(q,v)

vk − ∂K

∂qh

∣∣∣(q,v)

= ghk(q)wk + (∂kghj)q vjvk − 1

2(∂hgjk)q vjvk

= ghk(q)wk +

1

2(∂kghj)q vjvk +

1

2(∂jghk)q vjvk − 1

2(∂hgjk)q vjvk

= ghk(q)wk +

1

2(∂kghj + ∂jgkh − ∂hgjk)q vjvk

whence (∇γ

dt

)i

= (g−1 ◦ ([K] ◦ γ))i

= (g−1p ([K](p,v,w)))i

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

= wi +

{i

jk

}q

vjvk

Therefore, identities 8

(∇γ

dt

)i

= (γ − ΓK ◦ γ)i

= γi − (ΓK ◦ γ)i

read

wi +

{i

jk

}q

vjvk = wi − ΓiK(q, v)

Hence we obtain

ΓiK(q, v) = −

{i

jk

}q

vjvk

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

�

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


2.3 Euler-Lagrange equation

A 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 energy

V : 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 : TQ → R : (p,v) 7→ 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) ∈ TTQ | u = v , [K](p,v,w) = −dpV }

the mechanical energy E 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)

=d

dt(1

2mv · v) +

d

dtV (p)

= mw · v +d

dtV (p)

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

Hence our claim.�


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 : TQ → R : (p,v) 7→ 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 : T 2Q → T ∗Q : (p,v;v,w) 7→ (p, [L](p,v,w))

with[L](p,v,w) := [K](p,v,w) + dpV ∈ T ∗

pQ

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

DLagr = DEul−Lagr

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

DEul−Lagr := {(p,v;u,w) ∈ TTQ | 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 TQ is an integral curve of DEul−Lagr ,iff

(τQ ◦ c)� = c , [L] ◦ c = 0 (♦)

Proof The same as the proof, part (i), of Prop.8.

�

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

[L] ◦ γ = 0 (♦)′

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

�


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 γ : t ∈ I 7→ p = ξ(q(t)) ∈ U ⊂ Q – livingin the coordinate domain U of a chart ξ of Q – is a base integral curve ofDEul−Lagr , 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)

− ∂L∂qh

∣∣∣(q,v)

= 0 (♦)h

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

([L] ◦ γ)h = 0 (♦)′h

The above components are given by

([L] ◦ γ)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)

− ∂K

∂qh

∣∣∣(q,v)

+∂V

∂qh

∣∣∣q

=d

dt

∂(K − V )

∂vh

∣∣∣(q,v)

− ∂(K − V )

∂qh

∣∣∣(q,v)

=d

dt

∂L∂vh

∣∣∣(q,v)

− ∂L∂qh

∣∣∣(q,v)

where we have put v = q .

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

�


Remark For both L := K and L := K − V , the semi-basic 1-form [L] :T 2Q → T ∗Q is a mapping which takes any (p,v;v,w) ∈ T 2Q to the covector(p, [L](p,v,w)) ∈ T ∗Q characterized, in a chart, by components

([L](p,v,w))h =∂2L

∂vh∂vk

∣∣∣(q,v)

wk +∂2L

∂vh∂qk

∣∣∣(q,v)

vk − ∂L∂qh

∣∣∣(q,v)

Such a coordinate technique can be used to define [L] for an arbitrary La-grangian function L : S ⊂ TQ → R (smooth on an open subset S of TQ ),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 of‘variational principles’, as will now be shown.

Variational calculus

Let

γ : I → Q : t 7→ p(t)

be a smooth curve of Q .A smooth variaton of γ with fixed end-points in a closed sub-interval of I,

is a smooth mapping

χ : (−ε, ε)× J → Q : (s, t) 7→ p(s, t)

(with ε > 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 ∈ (−ε, ε)

χ can be thought of as a one-parameter family

{χs : J → Q : t 7→ ps(t) := p(s, t) }s∈(−ε,ε)

9 Actually, through higher geometric methods, [L] can be given a ‘coordinate-free’ defi-nition.


of ‘varied’ curves near γ|J = χo with fixed end-points in [t1, t2] ⊂ J ⊂ I ,whose tangent lifts { χs }s∈(−ε,ε) are all included in

χ : (−ε, ε)× J → TQ : (s, t) 7→(

p(s, t),∂p

∂t

∣∣∣(s,t)

)χ also define a one-parameter family

{χt : (−ε, ε) → Q : s 7→ pt(s) := p(s, t) }t∈J

of ‘isocronous’ curves, whose tangent vectors at the points of γ|J are the valuesof

χ′o : J → TQ : t 7→(

p(t),∂p

∂s

∣∣∣(0,t)

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

the integral 10

Iγ :=

∫ t2

t1

(L ◦ γ) dt

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

Iχ :=

∫ t2

t1

(L ◦ χ) dt : (−ε, ε) → R : s 7→ Iχs :=

∫ t2

t1

(L ◦ χs) dt

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

δγIχ :=dIχ

ds

∣∣∣0

is called the first variation of Iχ at γ .

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

δγIχ = −∫ t2

t1

〈[L] ◦ γ | χ′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).


Proof First remark that

δγIχ :=d

ds

∣∣∣0

∫ t2

t1

(L ◦ χ) dt =

∫ t2

t1

∂(L ◦ χ)

∂s

∣∣∣s=0

dt

Now, for any given to ∈ J , consider a chart ξ on a neighbourhood U of thepoint p(to) ∈ Im γ . In a suitably small neighbourhood of (0, to) ∈ (−ε, ε)×J ,χ takes values in U (by continuity) and will then have a local coordinate ex-pression (qh(s, t)). As a consequence, χ 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 χ′o will have components χ′oh := ∂qh

∂s

∣∣∣s=0

. With the aid of ξ , we obtain

∂(L ◦ χ)

∂s

∣∣∣(0,to)

=∂L∂qh

∣∣∣(q(to),v(to))

∂qh

∂s

∣∣∣(0,to)

+∂L∂vh

∣∣∣(q(to),v(to))

∂2qh

∂s∂t

∣∣∣(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))

− d

dt

∣∣∣to

∂L∂vh

∣∣∣(q,v)

)χ′o

h(to) +d

dt

∣∣∣to

(∂L∂vh

∣∣∣(q,v)

χ′oh

)

that is, 11

∂(L ◦ χ)

∂s

∣∣∣(0,to)

= −〈[L] ◦ γ | χ′o〉(to) +d

dt

∣∣∣to〈FL ◦ γ | χ′o〉

So, on the whole J , we have

∂(L ◦ χ)

∂s

∣∣∣s=0

= −〈[L] ◦ γ | χ′o〉+d

dt〈FL ◦ γ | χ′o〉

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

t1

d

dt〈FL ◦ γ | χ′o〉 dt = 〈FL ◦ γ | χ′o〉(t2)− 〈FL ◦ γ | χ′o〉(t1) = 0

Hence our claim.�

11 FL ◦ γ : I → T ∗Q will denote the (invariant) covector-valued map whose componentsin ξ are given by

(FL ◦ γ)h =∂L∂vh

∣∣∣(q,v)

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


Variational principle

A smooth curve γ : I → Q is said to be a geodesic curve of (Q, L) , if it satisfiesHamilton’s variational principle of stationary action, owing to which, for everysmooth variation χ of γ with fixed end-points in a closed sub-interval of I,Iχ is required to be stationary at γ, that is to say,

δγIχ = 0 , ∀χ

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

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

Proof (i) If γ : I → Q is a base integral curve of DEul−Lagr , i.e.

[L] ◦ γ = 0

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

t1

〈[L] ◦ γ | χ′o〉 dt = 0

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

(ii) Conversely, if γ : t ∈ I 7→ p(t) ∈ Q is not a base integral curve ofDEul−Lagr , say

([L] ◦ γ)(to) 6= 0 , to ∈ I

we shall prove that, for a suitable smooth variation χ of γ with fixed end-points in a closed sub-interval [t1, t2] of I ,∫ t2

t1

〈[L] ◦ γ | χ′o〉 dt 6= 0

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

To this end, consider a chart on a neighbourhood

U 3 p(to)

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

([L] ◦ γ)1(to) > 0


By continuity, there exists a suitably small open interval J ⊂ I containing tos.t., for all t ∈ J ,

p(t) ∈ U , ([L] ◦ γ)1(t) > 0

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

q1(s, t) := q1(t) + s(cos(t− to)− cosα)

q2(s, t) := q2(t)

. . . . . .

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

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

0 < α <π

2s.t. [to − α, to + α] ⊂ J

By doing so, we define a map

χ : (−ε, ε)× J → Q : (s, t) 7→ p(s, t)

which is immediately seen to be a smooth variation of γ with fixed end-pointsin

[t1, t2] := [to − α, to + α]

The components of χ′o are , for all t ∈ J ,

χ′o1(t) = cos(t− to)− cosα

χ′o2(t) = 0

. . . . . .

χ′on(t) = 0

and, for all t ∈ (t1, t2) ,

χ′o1(t) > 0

Hence∫ t2

t1

〈[L] ◦ γ | χ′o〉 dt =

∫ t2

t1

([L] ◦ γ)h χ′oh dt =

∫ t2

t1

([L] ◦ γ)1 χ′o1 dt > 0

which is our claim.

�


Riemannian case

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

[K] ◦ γ = 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 ◦ γ = const. = 0

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

K ◦ γ = const. > 0

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

Let us consider one more Lagrangian function, namely

Λ :=√

2K

which is clearly smooth on TQ \K−1(0) . 12

If γ : t ∈ I → Q : t 7→ p(t) ∈ Q is a smooth curve satisfying

Im γ ⊂ TQ \K−1(0)

(i.e. p(t) 6= 0 for all t ∈ I ) and [t1, t2] ⊂ I , the integral

Lγ :=

∫ t2

t1

(Λ ◦ γ) dt

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

12 Λ is smooth only on TQ \K−1(0) , since, in any chart, its partial derivatives

∂Λ∂qh

∣∣∣(q,v)

=1√

2K(q, v)∂K

∂qh

∣∣∣(q,v)

,∂Λ∂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) ∈ K−1(0).13 For any dt > 0 , Λ(p(t), p(t)) dt =

√〈gp(t)(dp) | dp〉 > 0 defines the length, in the

given Riemannian metric, of the ‘infinitesimal arc’ dp := p(t)dt .


The variational theory previously sketched, applied to Λ , 14 shows that γis a curve of stationary length, that is,

δγLχ = 0 , ∀χ

iff[Λ] ◦ γ = 0

Proposition 17 γ 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 γ is a uniform motion, that is,

K ◦ γ = κ = const. > 0

then

[Λ] ◦ γ =1√2κ

[K] ◦ γ

In order to prove the lemma, recall that, along the uniform motion γ : t ∈I 7→ p(t) ∈ Q , we have (for any given t ∈ I and any chart at p(t) ) 15

([Λ] ◦ γ)h =d

dt

∂Λ

∂vh

∣∣∣(q,v)

− ∂Λ

∂qh

∣∣∣(q,v)

=d

dt

(1√2κ

∂K

∂vh

∣∣∣(q,v)

)− 1√

2κ

∂K

∂qh

∣∣∣(q,v)

=1√2κ

(d

dt

∂K

∂vh

∣∣∣(q,v)

− ∂K

∂qh

∣∣∣(q,v)

)=

1√2κ

([K] ◦ γ)h

which is our claim.�

14 See footnote 10.15 See footnote 12.

Chapter 3

From Lagrange to Hamilton

The Lagrangian dynamics of a conservative system S –whose geometricalarena is the velocity phase space TQ , supporting Euler-Lagrange equation –will now be taken into the range of Hamiltonian dynamics, where the geo-metrical arena is the momentum phase space T ∗Q , supporting a ‘Hamiltonequation’ which, defined in terms of the canonical symplectic structure of T ∗Q,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) ∈ TQ onto the corre-sponding momentum (p, mv · |TpQ) ∈ T ∗Q .

Lagrangian function and Legendre transformation

The above map is indeed the vector bundle isomorphism

g : TQ → T ∗Q : (p,v) 7→ (p, gp(v)) , gp(v) := mv · |TpQ

owing to which the configuration space of the system has been given the struc-ture of a Riemannian manifold (Q,K) . 1

Such 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 : TQ → T ∗Q : (p,v) 7→ (p, FpL(v)) , FpL(v) := gp(v)

(whence πQ ◦ FL = τQ ).

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

33

34 3 From Lagrange to Hamilton

Coordinate formalism

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

Q , and the components p = (p1, . . . , pn) of π in ξ , given by ph = 〈π | ∂p∂qh

∣∣∣q〉 .

In natural coordinate formalism, Legendre transformation

(p,v) 7→ (p, π) , π = FpL(v)

is expressed by 3

(q, v) 7→ (q, p) , ph = ghk(q) vk =∂L∂vh

∣∣∣(q,v)

(1)

The inverse transformation

(p, π) 7→ (p,v) , v = (FpL)−1(π)

is in turn expressed by

(q, p) 7→ (q, v) , vh = ghk(q) pk =: νh(q, p) (2)

Remark Such a coordinate technique can be used to define a Legendremorphism FL for an arbitrary Lagrangian function L : S ⊂ TQ → T ∗Q(smooth on an open subset S of TQ ), since – also in such case – the valueπ = FpL(v) ∈ T ∗

pQ (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) | ∂p

∂qh

∣∣∣q

⟩=

⟨gp

( ∂p∂qh

∣∣∣q

)| v

⟩=

⟨gξ(q)

( ∂p∂qh

∣∣∣q

)| ∂p

∂qk

∣∣∣q

⟩vk = ghk(q) vk =

∂K

∂vh

∣∣∣(q,v)

=∂L∂vh

∣∣∣(q,v)

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

3 From Lagrange to Hamilton 35

3.2 Hamiltonian function

Legendre transformation also determines transition from the Lagrangian func-tion (on TQ ) to a ‘Hamiltonian function’ (on T ∗Q ) .

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 : TQ → R

The Hamiltonian function

H := E ◦ (FL)−1 : T ∗Q → R

is the ‘push forward’ of E by Legendre transformation FL .


In natural coordinate formalism, the Hamiltonian function

(p, π) 7→ H(p, π) = E((FL)−1(p, π)) = E(p, (FpL)−1(π)) = E(p,v) = 〈FpL(v) | v〉 − L(p,v)

= 〈π | v〉 − L(p,v) , v = (FpL)−1(π)

is expressed by

(q, p) 7→ H(q, p) = pk vk − L(q, v) , v = ν(q, p)

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

∂H

∂qh

∣∣∣(q,p)

= pk∂vk

∂qh

∣∣∣(q,p)

− ∂L∂vk

∣∣∣(q,v)

∂vk

∂qh

∣∣∣(q,p)

− ∂L∂qh

∣∣∣(q,v)

= − ∂L∂qh

∣∣∣(q,v)

(3)

and∂H

∂ph

∣∣∣(q,p)

= vh + pk∂vk

∂ph

∣∣∣(q,p)

− ∂L∂vk

∣∣∣(q,v)

∂vk

∂ph

∣∣∣(q,p)

= νh(q, p) (4)

5 We put

〈FL | idTQ〉 : (p,v) ∈ TQ 7−→ 〈FpL(v) | idTQ(v)〉 = 〈FpL(v) | v〉 = 〈gp(v) | v〉 = 2K(p,v) ∈ R


3.3 Hamiltonian vector field

The canonical symplectic geometry of T ∗Q will now be taken into considera-tion.

Canonical symplectic structure and Hamiltonian vector field

Letω : TM → T ∗M : (π, X) 7→ (π, ωπ(X))

be the canonical symplectic structure of cotangent bundle M := T ∗Q . 6

As ω is a vector bundle isomorphism of TM onto T ∗M , it transformsthe differential

dH : M → T ∗M

of Hamiltonian function H into the Hamiltonian vector field

XH := ω−1 ◦ dH : M → TM


Recall that, on the 2n-dimensional manifold M = T ∗Q , the canonical sym-plectic structure ω is characterized, in any natural chart, by the 2n × 2nmatrix of components [

ω(1)(1) ω(1)(2)

ω(2)(1) ω(2)(2)

]with ω(1)(1) = ω(2)(2) = 0 (n×n zero matrix) and ω(1)(2) = −ω(2)(1) = δ (n×nidentity matrix).

If, for any π ∈ M , [dπH(1) dπH(2)

]and [

XH(π)(1)

XH(π)(2)

]are the 2n-tuples of components of dπH ∈ T ∗

πM and XH(π) ∈ TπM , respec-tively, the linear mapping

dπH = ωπ(XH(π))

is expressed by 7

dπH(1) = ω(α)(1)XH(π)(α) , dπH(2) = ω(α)(2)XH(π)(α)

6 Any ‘point’ (p, π) ∈ T ∗Q is now referred to as π ∈ M .7 Summation over (α) = (1), (2) is understood.

3 From Lagrange to Hamilton 37

that is,dπH(1) = −XH(π)(2) , dπH(2) = XH(π)(1)

As

dπH(1) =

[∂H

∂qh

∣∣∣(q,p)

], dπH(2) =

[∂H

∂ph

∣∣∣(q,p)

]we obtain

XH(π)(1) =

[∂H

∂ph

∣∣∣(q,p)

], XH(π)(2) =

[−∂H

∂qh

∣∣∣(q,p)

](5)

3.4 Hamilton equation

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

Cotangent dynamically possible motions

If c is an integral curve of DEul−Lagr , 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, TDPM s and CDPM s bijectively correspondto one another. Through such a bijection, the problem of determining theTDPM s proves to be naturally equivalent to that of determining the CDPM s.

Hamilton equation

Proposition 18 The CDPMs are the integral curves of Hamilton equation

DHam := Im XH

Proof Letk : I → T ∗Q , c : I → TQ

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

k = FL ◦ c

and then both projecting down onto the same curve

πQ ◦ k = πQ ◦ FL ◦ c = τQ ◦ c : I → Q

We shall prove that

(♦) (τQ ◦ c)� = c , [L] ◦ c = 0 ⇐⇒ k = XH ◦ k (�)


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 = νh(q, p).

So condition (♦), that is,

qh = vh ,d

dt

∂L∂vh

∣∣∣(q,v)

− ∂L∂qh

∣∣∣(q,v)

= 0

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

qh = νh(q, p) , ph =∂L∂qh

∣∣∣(q,v)

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

qh =∂H

∂ph

∣∣∣(q,p)

, ph = −∂H

∂qh

∣∣∣(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 πQ ◦ k , i.e.

k = FL ◦ c = FL ◦ (τQ ◦ c)� = FL ◦ (πQ ◦ k)�

The integral curves and the base integral curves of DHam are then bi-jectively related to one another by projection πQ and, owing to the abovebijection, the first-order problem of determining the integral curves of DHam

proves to be naturally equivalent to the second-order problem of determiningits base integral curves, i.e. the solution curves γ of

(FL ◦ γ)� = XH ◦ (FL ◦ γ)

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 that[

qh

ph

]are the components of k .

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) ∈ TTQ | u = v , [K](p,v,w) = F (p,v)}

is then the dynamics of M , the DPM s of M being defined as the baseintegral curves γ of DLagr and hence characterized by condition

[K] ◦ γ = F ◦ γ

If F = 0 , the DPM s γ of M – characterized by a Riemannian geodesiccurvature [K] ◦ γ identically vanishing – coincide with the geodesic curves ofRiemannian manifold (Q, K) , which can therefore be called inertial motionsin (Q,K) .

If F 6= 0 , the DPM s γ of M – characterized by a Riemannian geodesiccurvature [K] ◦ γ differing from zero as much as is imposed by the covectorforce F ◦ γ – 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

40 4 Concluding remarks

4.2 Gauge transformations

However 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 7→ L := K − V , F 7→ F := F + dV

On the one hand, such a transformation gives rise to a new mechanicalsystem

M := (Q, L, F)

with Lagrange equation

DLagr = {(p,v;v,w) ∈ T 2Q | u = v , [L](p,v,w) = F(p,v)}

and DPM s – the base integral curves of DLagr – characterized by condition

[L] ◦ γ = F ◦ γ

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

If F 6= 0 , the DPM s γ of M – characterized by a Lagrangian geodesiccurvature [L] ◦ γ differing from zero as much as is imposed by the covectorforce F ◦ γ – 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) ∈ T 2Q , it follows that

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

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

4 Concluding remarks 41

whenceDLagr = DLagr

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

4.3 Geometrizing physical fields

From the physical point of view, the meaning of the above kind of gaugetransformations 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 ofthe particle system into a ‘perturbed’ Lagrangian geometry L = K − V .

That anticipates, in a sense, Einstein’s 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 onan open subset of TQ ).

The dynamics of M will then be defined by the corresponding Lagrangeequation 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.

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

5 Appendix 43

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

A = Im ξ

with 4

ξ : Rn → E : q 7→ p = ξ(q) = o + Ξ(q) = o + qheh

(where o ∈ A and Ξ : v ∈ Rn 7→ Ξ(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) ξ : Rn → A is a homeomorphism; 5

(2) ξ : Rn → E is C∞- differentiable, with injective differential dqξ = Ξat 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 ξ

of a local parametrization

ξ : W ⊂ Rn → E : q 7−→ p = ξ(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 ξ is C∞- differentiable, iff it is continuous and admits

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

dqξ : Rn → E : v = (vh) 7−→ v = vh ∂p∂qh

∣∣∣q

44 5 Appendix

1. ξ : W → U is a homeomorphism;

2. ξ : W → E is C∞- differentiable, with injective differential dqξ at eachq ∈ W .

Remark that dqξ is injective, iff its image

Im dqξ = Span

(∂p

∂qh

∣∣∣q

)(spanned by the first-order partial derivatives of ξ ) is an n-dimensional vectorsubspace 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 ξ . Each pointp ∈ U is then given a unique n-tuple of coordinates q = ξ−1(p) ∈ W by theinverse map ξ−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 U being 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 space’at 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 ‘derivation’as follows.

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

5 Appendix 45

Consider a smooth curve of E , i.e. a C∞- differentiable E-valued map

γ : I → E : t 7→ γ(t) = p(t)

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

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

Tangent vector spaces

Now consider the smooth curves γ of E with orbits lying on Q, i.e.

Im γ ⊂ Q

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

p ∈ Im γ

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 ⊂ E

of 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 ξ : W → U at p (i.e. U 3 p = ξ(q) , with q ∈ W ),

TpQ = Im dqξ (†)

(i) Let v ∈ TpQ , i.e.

v = p(to)

for some smooth curve γ : t ∈ I 7→ p(t) ∈ Q through p(to) = p (with to ∈ I ).

8 Think of the graphic representation of derivative p(t) := lim∆t→01

∆t

(p(t+∆t)−p(t)

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

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. γ(Io) ⊂ U . As a consequence, γ cabe given the local C∞ coordinate expression

α = ξ−1 ◦ γ|Io : Io → W : t 7→ q(t) = ξ−1(p(t))

By derivation at to of p(t) = ξ(q(t)) , we obtain

v = p(to)

= qh(to)∂p

∂qh

∣∣∣q(to)

= dq(to)ξ (q(to))

= dqξ (v) , v := q(to) ∈ Rn

Hence v ∈ Im dqξ .

(ii) Now let v ∈ Im dqξ , i.e.

v = dqξ (v) , v ∈ Rn

Choose to ∈ R and consider the affine map

α : R → Rn : t 7→ q(t) := q + (t− to)v

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. α(Io) ⊂ W . As a consequence,

γ = ξ ◦ α|Io : Io → U : t 7→ p(t) = ξ(q(t))

is a smooth curve of Q through

p(to) = ξ(q(to)) = ξ(q) = p

whose derivative at to is

p(to) = qh(to)∂p

∂qh

∣∣∣q(to)

= dq(to)ξ (q(to))

= dqξ (v)

= v

Hence v ∈ TpQ .�

Remark thatv ∈ TpQ 7−→ v = (dqξ)

−1(v) ∈ Rn

is the linear isomorphism which takes any vector v ∈ TpQ to its components

v ∈ Rn in the basis

(∂p∂qh

∣∣∣q

)of TpQ (called linear components of v in ξ ).

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, . . . , gµ) : E → Rµ : p 7−→ g(p) = (g1(p), . . . , gµ(p))

is a continuous map, then

U := g−1(R+)µ = {p ∈ E | g(p) ∈ (R+)µ}= {p ∈ E | g1(p) > 0, . . . , gµ(p) > 0}

(inverse image by g of the open subset (R+)µ ⊂ Rµ ) 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, . . . , fκ) : U ⊂ E → Rκ : p 7−→ f(p) = (f1(p), . . . , fκ(p))

be a differentiable map, defined on an open subset U ⊂ E (say U = g−1(R+)µ )and taking values in Rκ (κ < m := dim E ), with surjective differential dpf :E → Rκ at each point p of

Q := f−1(0) = {p ∈ U | f(p) = 0} = {p ∈ U | f1(p) = 0, . . . , fκ(p ) = 0}= {p ∈ E | f1(p) = 0, . . . , fκ(p ) = 0, g1(p) > 0, . . . , gµ(p) > 0}

Q is then an n-dimensional embedded submanifold of E with

n := m− κ

and, for all p ∈ Q ,TpQ = Ker dpf

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 κ-tuple of real-valued functionsϕ1 : W → R, . . . , ϕκ : W → R s.t.

U = Im ξ

where

ξ : W → E

is a map defined (with a suitable choice of affine coordinates φ : Rm → E ) by

(x1, . . . , xn)ξ7−→ φ( x1, . . . , xn, ϕ1(x1, . . . , xn), . . . , ϕκ(x1, . . . , xn) )

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

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

dpf ◦ dqξ = dq(f ◦ ξ) = 0

(with p = ξ(q) ) and then

Im dqξ ⊂ Ker dpf

Moreover, the surjectivity of dpf and the injectivity of dqξ imply the dimen-sional result

dim Ker dpf = dim E − dim Im dpf = m− κ = n = dim Im dqξ

So we obtain

Im dqξ = Ker dpf

which, owing to (†), completes our proof.

�

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

f1

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

)= 0, . . . , fκ

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

)= 0

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

xn+1 = ϕ1(x1, . . . , xn), . . . , xn+κ = ϕκ(x1, . . . , xn)

5 Appendix 49

5.2 Tangent bundle and differential equations

The tangent bundle of a manifold is the first geometric arena of differentialequations.

Tangent bundle and canonical projection

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

TQ =⋃p∈Q

{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 TQ onto its base manifold Q is the map

τQ : TQ → Q : (p,v) 7→ p

Vector fibre bundle structure

TQ 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 chartsξ : W → U of Q )

ξ1 : (q, v) ∈ W × Rn 7→ (p,v) = (ξ(q), dqξ (v)) ∈ τ−1Q (U)

which give TQ the structure of a 2n-dimensional smooth manifold embeddedin TE = E × E . 10

Tangent lift

The tangent lift of a smooth curve

γ : I → Q : t 7→ γ(t) = p(t)

living in Q ⊂ E , is the smooth curve

γ : I → TQ : t 7→ γ(t) = (p(t), p(t))

living in TQ ⊂ TE .

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

50 5 Appendix

Differential equations in implicit form

From the very definition of tangent vector spaces, it follows that TQ 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 ⊂ TQ

is assigned (by prescribing some geometric or algebraic property), the problemmay arise of determining the smooth curves γ of Q whose tangent lifts livein D , i.e.

Im γ ⊂ D (�)

that is to say,

γ(t) = (p(t), p(t)) ∈ D , ∀t ∈ I (�)

Such a D will be called (autonomous) first-order differential equation inimplicit form on Q . 11

Integral curves

Any solution to the above problem, i.e. any smooth curve γ 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) ⊂ D

swept by the tangent lifts of all its integral curves.

D is said to be integrable, if

∅ 6= D = D(i)

11 We shall not consider the case of a non-autonomous equation D ⊂ J1Q , subset ofthe jet bundle J1Q := R × TQ , whose solutions are the smooth curves γ of Q with jetextension j1γ : t ∈ I 7→ j1γ (t) := (t, p(t), p(t)) ∈ J1Q living in D , i.e. Im j1γ ⊂ D .

5 Appendix 51

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 → TQ : p 7→ (p, X(p))

on Q 12 s.t.

D = Im X = {(p,v) ∈ TQ | v = X(p)}

For such an equation, condition (�) reads

γ = X ◦ γ

that is to say,

p(t) = X(p(t)) , ∀t ∈ I

Cauchy problems and determinism

The integrability of

D = Im X 6= ∅

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 γ : t ∈ I 7→ p(t) ∈ Q of D , with I 3 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 γo ⊂ D(i)

whence

D = D(i)

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

q ∈ W 7−→ (dqξ)−1(X(ξ(q))

)∈ Rn

52 5 Appendix

5.3 Second tangent bundle and second-order

differential 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

TTQ ⊂ TTE

of TQ ⊂ TE .

Recall that TTQ is the region of TTE = (E ×E)× (E×E) swept by thetangent lifts of all the smooth curves of TQ , the tangent lift of any such curve

c : I → TQ : t 7→ c(t) = (p(t),v(t))

being given by

c : I → TTQ : t 7→ c(t) = (p(t),v(t); p(t), v(t))

In particular, the second tangent lift of a smooth curve

γ : I → Q : t 7→ γ(t) = p(t)

living in Q , is the tangent lift of

γ : I → TQ : t 7→ γ(t) = (p(t), p(t))

i.e. the smooth curve 13

γ : I → T 2Q : t 7→ γ(t) = (p(t), p(t); p(t), p(t))

living in the region T 2Q ⊂ TTQ , called second tangent bundle of Q, definedby

T 2Q = {(p,v;u,w) ∈ TTQ | u = v}

13 p(t) will denote the derivative of p(t) .

5 Appendix 53

Affine fibre bundle structure

Remark thatT 2Q =

⋃(p,v)∈TQ

{(p,v)} ×({v} × T 2

(p,v)Q)

withT 2

(p,v)Q := {w ∈ E | (p,v;v,w) ∈ TTQ}is an affine fibre bundle over TQ , in the sense that its fibre {(p,v)} ×({v} × T 2

(p,v)Q)

over each (p,v) ∈ TQ is an affine space (canonically iso-

morphic to T 2(p,v)Q ), as will now be shown.

Proposition 21 For any (p,v) ∈ TQ , T 2(p,v)Q is an n-dimensional affine

subspace of E modelled on TpQ , i.e.

T 2(p,v)Q = z + TpQ

for some z ∈ E .

Proof Follows from property (†), owing to the followingLemma For any natural chart ξ1 : W×Rn → τ−1

Q (U) at (p,v) (i.e. τ−1Q (U) 3

(p,v) = (ξ(q), dqξ (v)) with (q, v) ∈ W × Rn ),

T 2(p,v)Q = z(q, v) + Im dqξ (††)

where z(q, v) := vhvk ∂2p∂qh∂qk

∣∣∣q.

(i) Let w ∈ T 2(p,v)Q , that is,

(p,v;v,w) ∈ TTQ

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

for some smooth curve c : t ∈ I 7→ c(t) = (p(t),v(t)) ∈ TQ and to ∈ I .As U is 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|Io

can be given the C∞ coordinate expression

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

∂qh

∣∣∣q(t)

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

54 5 Appendix

By derivation at to , we obtain

w = v(to)

= vh(to)qk(to)

∂2p

∂qh∂qk

∣∣∣q(to)

+ vh(to)∂p

∂qh

∣∣∣q(to)

= vhvk ∂2p

∂qh∂qk

∣∣∣q+ wh ∂p

∂qh

∣∣∣q, w := v(to)

= z(q, v) + dqξ (w)

Hence w ∈ z(q, v) + Im dqξ .

(ii) Now let w ∈ z(q, v) + Im dqξ , that is,

w=z(q, v) + dqξ (w)

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

α : R → Rn : t 7→ q(t) := q + v(t− to) +1

2w(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. α(Io) ⊂ W . As a consequence,

γ = ξ ◦ α|Io : Io → U : t 7→ p(t) = ξ(q(t))

is a smooth curve of Q through

p(to) = ξ(q(to)) = ξ(q) = p

whose derivatives at to are

p(to) = qh(to)∂p

∂qh

∣∣∣q(to)

= dq(to)ξ (q(to))

= dqξ (v)

= v

and

p(to) = qh(to)qk(to)

∂2p

∂qh∂qk

∣∣∣q(to)

+ qh(to)∂p

∂qh

∣∣∣q(to)

= vhvk ∂2p

∂qh∂qk

∣∣∣q+ wh ∂p

∂qh

∣∣∣q

= z(q, v) + dqξ (w)

= w

5 Appendix 55

So

(p,v;v,w) = (p(to), p(to); p(to), p(to)) = γ(to) ∈ Im γ ⊂ T 2Q ⊂ TTQ

Hence w ∈ T 2(p,v)Q .

�

Remark that

w ∈ T 2(p,v)Q 7−→ w = (dqξ)

−1(w − z(q, v)) ∈ Rn

is the affine isomorphism which takes any vector w ∈ T 2(p,v)Q to its affine

coordinates w ∈ Rn in the frame

(z(q, v),

(∂p∂qh

∣∣∣q

))of T 2

(p,v)Q (called affine

components of w in ξ ).

Second-order differential equations in implicit form

Part (ii) of the above proof shows as well that T 2Q is the region of TTQ‘swept’ by the second tangent lifts of all the smooth curves of Q .

As a consequence, if a subset

D ⊂ T 2Q

is assigned (by prescribing some geometric or algebraic property), the problemmay arise of determining the smooth curves γ : t ∈ I 7→ p(t) ∈ Q whosesecond tangent lifts live in D , i.e.

Im γ ⊂ D (��)

that is to say,

γ(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 .

56 5 Appendix

Integral curves and base integral curves

A second-order differential equation on Q , say D ⊂ T 2Q ⊂ TTQ , is a first-order differential equation on TQ as well, whose integral curves exhibit thefollowing peculiar property.

Proposition 22 Each integral curve c of D ⊂ T 2Q is the tangent lift of itsown projection τQ ◦ c , i.e.

(τQ ◦ c)� = c

Proof Letc : t ∈ I 7→ c(t) = (p(t),v(t)) ∈ TQ

be a smooth curve of TQ , and

τQ ◦ c : t ∈ I 7→ τQ(c(t)) = p(t) ∈ Q

its 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)) ∈ D

whencep(t) = v(t)

that is,(τQ ◦ c)�(t) = (p(t), p(t)) = (p(t),v(t)) = c(t)

which is our claim.�

The base integral curves γ = τQ◦c 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 γ of Q is a base integral curve of D , iff itis a solution curve of problem (��).

Proof (i) Let Im γ ⊂ D . Putting c := γ , whence τQ ◦ c = τQ ◦ γ and c = γ ,we obtain τQ ◦ c = γ and Im c ⊂ D .(ii) Conversely, let γ = τQ ◦ c and Im c ⊂ D . From Prop. 22, we obtainγ = (τQ ◦ c)� = c , whence γ = c , and then Im γ ⊂ D .

�

The integral curves and the base integral curves of D are then bijectivelyrelated to one another by projection τQ (Prop. 22) and, owing to the above bi-jection, the first-order problem of determining the integral curves of D provesto be naturally equivalent to the second-order problem of determining its baseintegral curves, i.e. the solution curves γ of (��) (Prop. 23).

5 Appendix 57

Semi-sprays and normal form

A second-order equation D is then reducible to normal form, if there exists asmooth semi-spray

Γ : TQ → T 2Q ⊂ TTQ : (p,v) 7→ (p,v;v, Γ(p,v))

on TQ 14 s.t.

D = Im Γ = {(p,v;u,w) ∈ TTQ | u = v , w = Γ(p,v)}Proposition 24 Condition Im c ⊂ D , characterizing the integral curves c ofD = Im Γ , reads

c = Γ ◦ c

Condition Im γ ⊂ D , characterizing the base integral curves γ of D = Im Γ ,reads

γ = Γ ◦ γ

Proof (i) For a smooth curve c : t ∈ I 7→ c(t) = (p(t),v(t)) ∈ TQ , condition

c(t) = (p(t),v(t); p(t), v(t)) ∈ D = Im Γ , ∀t ∈ I

readsp(t) = v(t) , v(t) = Γ(p(t),v(t)) , ∀t ∈ I

that is,

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

= (p(t),v(t);v(t), Γ(p(t),v(t)))

= (Γ ◦ c) (t) , ∀t ∈ I

(ii) For a smooth curve γ : t ∈ I 7→ p(t) ∈ Q , condition

γ(t) = (p(t), p(t); p(t), p(t)) ∈ D = Im Γ , ∀t ∈ I

readsp(t) = Γ(p(t), p(t)) , ∀t ∈ I

that is,

γ(t) = (p(t), p(t); p(t), p(t))

= (p(t), p(t); p(t), Γ(p(t), p(t)))

= (Γ ◦ γ) (t) , ∀t ∈ I

�14 Γ (vector field on TQ taking values in T 2Q ) is said to be smooth, if – in any chart

ξ : W → U – it admits C∞ affine components

(q, v) ∈ W × Rn 7−→ (dqξ)−1(Γ(ξ(q), dq ξ(v))− z(q, v)

)∈ Rn

58 5 Appendix

Cauchy problems and determinism

If D = Im Γ , then – owing to determinism theorem– Cauchy problem (D, to, (po,vo)) ,for any to ∈ R and (po,vo) ∈ TQ , admits a unique maximal solution, whichamounts to saying that there exists a unique maximal base integral curveγ : t ∈ I 7→ p(t) ∈ Q of D , with to ∈ I , satisfying initial conditions(p(to), p(to)) = (po,vo) (all of the other solutions to the problem being restric-tions of the maximal one).

5.4 Cotangent bundle and differential forms

The cotangent bundle of a manifold is the geometric arena of differential forms.

Cotangent vector spaces

For any p ∈ Q , to the tangent vector space TpQ there corresponds, by duality,the cotangent vector space T ∗

pQ made up of all the covectors

π : TpQ → R : v 7→ 〈π | v〉

(linear maps of TpQ in R ).Covectors can naturally be summed to one another and multiplied by

scalars, so that T ∗pQ turns into a vector space.

To the basis

(∂p∂qh

∣∣∣q

)determined in TpQ by a chart ξ at p = ξ(q) , there

corresponds a dual basis (dpqh) in T ∗

pQ , defined by

dpqh : TpQ → R : v = vk ∂p

∂qk

∣∣∣q7−→ 〈dpq

h | v〉 := vh

The components (ph) which characterize any covector π ∈ T ∗pQ in the

above basis – called components of π in ξ – are given by

ph :=

⟨π | ∂p

∂qh

∣∣∣q

⟩since, by pairing π with all v ∈ TpQ , we obtain

〈π | v〉 =

⟨π | vh ∂p

∂qh

∣∣∣q

⟩=

⟨π | ∂p

∂qh

∣∣∣q

⟩vh = phv

h = ph〈dpqh | v〉 = 〈phdpq

h | v〉

that is,π = ph dpq

h

5 Appendix 59

Cotangent bundle and canonical projection

The cotangent bundle of an n-dimensional smooth manifold Q is the disjointunion of its cotangent vector spaces, i.e.

T ∗Q :=⋃p∈Q

{p} × T ∗pQ

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

πQ : T ∗Q → Q : (p, π) 7→ p

Vector bundle structure

T ∗Q is a vector fibre bundle over Q , since its fibre {p} × T ∗pQ over each

p ∈ Q is a vector space (canonically isomorphic to T ∗pQ ).

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

ξ1 : (q, p) ∈ W × Rn 7→ (p, π) = (ξ(q), ph dpqh) ∈ π−1

Q (U)

which give T ∗Q the structure of a 2n-dimensional smooth manifold. 15

Differential 1-forms

A smooth real-valued function V : Q → R 16 gives rise to a covector field

dV : Q → T ∗Q : p 7→ (p, dpV )

called differential of V or exact 1-form on Q , where the differential dpV ∈T ∗

pQ of V at p is defined by putting, for any vector v = p(to) ∈ TpQ tangentto a smooth curve γ : t ∈ I 7→ p(t) ∈ Q at p = p(to) ,

〈dpV | v〉 :=d

dt(V ◦ γ)

∣∣∣to

15 Unlike TQ ⊂ TE = E × E , cotangent bundle T ∗Q is not naturally embedded insome Euclidean space related to E , and therefore its structure of smooth manifold is to bemeant in the generalized sense explained in the last subsection ‘Intrinsic geometry of smoothmanifolds’ of this Appendix.

16 V is said to be smooth, if – in any chart ξ : W → U – it admits a C∞ coordinateexpression

Vξ := V ◦ ξ : q ∈ W 7→ V (ξ(q)) ∈ R

60 5 Appendix

and then, by suitably restricting γ to the coordinate domain of a chart ξ wherep = ξ(q) andv = dqξ(v) ,

〈dpV | v〉 =d

dt(Vξ ◦ ξ−1 ◦ γ)

∣∣∣to

=∂Vξ

∂qh

∣∣∣q(to)

qh(to)

=∂Vξ

∂qh

∣∣∣qvh

=∂Vξ

∂qh

∣∣∣q〈dpq

h | v〉

=

⟨∂Vξ

∂qh

∣∣∣qdpq

h | v⟩

that is to say,

dpV =∂V

∂qh

∣∣∣pdpq

h

with∂V

∂qh

∣∣∣p

:=∂Vξ

∂qh

∣∣∣q

The above considerations can all be extended to real-valued functions de-fined on open subsets of Q (check, for instance, that dpq

k is the differential atp of the k-th projection qk : U → R of ξ−1).

More generally, a differential 1-form on Q is defined as a smooth covectorfield

f : Q → T ∗Q : p 7→ (p, f(p))

Its value f(p) ∈ T ∗pQ is expressed in a chart ξ at p = ξ(q) by

f(p) = fh(q) dpqh

in terms of components 17

fh(q) :=

⟨f(p) | ∂p

∂qh

∣∣∣q

⟩17 f is said to be smooth, if – in any chart ξ : W → U – it admits C∞ components

fh : q ∈ W 7→ fh(q) ∈ R

5 Appendix 61

Semi-basic differential 1-forms

The concept of 1-form furtherly generalizes into that of semi-basic 1-form onTQ , defined as a smooth bundle morphism

F : TQ → T ∗Q : (p,v) 7→ (p, F (p,v))

Its value F (p,v) ∈ T ∗pQ is expressed in a chart ξ –where p = ξ(q) and

v = dqξ(v) – by

F (p,v) = Fh(q, v) dpqh


Fh(q, v) :=

⟨F (p,v) | ∂p

∂qh

∣∣∣q

⟩The concept of semi-basic 1-form on T 2Q is defined in the same way, as a

smooth bundle morphism

Φ : T 2Q → T ∗Q : (p,v;v,w) 7→ (p, Φ(p,v,w))

Its value Φ(p,v,w) ∈ T ∗pQ is expressed in a chart ξ –where p = ξ(q) ,

v = dqξ(v) and w = z(q, v) + dqξ (w) – by

Φ(p,v,w) = Φh(q, v, w) dpqh


Φh(q, v, w) =

⟨Φ(p,v,w) | ∂p

∂qh

∣∣∣q

⟩Special kinds of semibasic 1-forms will now be described.

18 F is said to be smooth, if – in any chart ξ : W → U – it admits C∞ components

Fh : (q, v) ∈ W × Rn 7→ Fh(q, v) ∈ R

19 Φ is said to be smooth, if – in any chart ξ : W → U – it admits C∞ components

Φh : (q, v, w) ∈ W × Rn × Rn 7→ Φh(q, v, w) ∈ R

62 5 Appendix

Semi-Riemannian metric

A semi-Riemannian metric on an n-dimensional manifold Q is a symmetricvector bundle isomorphism of TQ onto T ∗Q , that is, a semi-basic 1-form

g : TQ → T ∗Q : (p,v) 7→ (p, gp(v))

such that, for each p ∈ Q , its restriction

gp : TpQ → T ∗pQ : v 7→ gp(v)

is a linear isomorphism (non-degenerateness) and the bilinear inner product〈gp(u) | v〉 of any two vectors u,v ∈ TpQ satisfies the symmetry condition(commutativity)

〈gp(u) | v〉 = 〈gp(v) | u〉

If non-degenerateness is strengthened by requiring positive definiteness, i.e.〈gp(v) | v〉 > 0 for all (p,v) ∈ TQ with v 6= 0 , g is said to be a Riemannianmetric.

Owing to polarization identity

〈gp(u) | v〉 =1

2

(〈gp(u + v) | u + v〉 − 〈gp(u) | u〉 − 〈gp(v) | v〉

)(for all p ∈ Q and u,v ∈ TpQ ), g is characterized by its quadratic form

K : TQ → R : (p,v) 7→ K(p,v) :=1

2〈gp(v) | v〉

A manifold equipped with a (semi-)Riemannian metric will then be denotedby (Q, K) and called (semi -)Riemannian manifold.

Remark that, any semi-spray

ΓK : TQ → T 2Q : (p,v) 7→ (p,v;v, ΓK(p,v))

associated with a (semi-)Riemannian manifold (Q,K) , naturally transforms

g : TQ → T ∗Q : (p,v) 7→ (p, gp(v))

(semi-basic 1-form on TQ ) into

[K] : T 2Q → T ∗Q : (p,v;v,w) 7→ (p, [K](p,v,w))

(semi-basic 1-form on T 2Q ) by putting

[K](p,v,w) := gp(w − ΓK(p,v)) ∈ T ∗pQ

5 Appendix 63

In a chart ξ at p = ξ(q) , the inner product 〈 gp(u) | v 〉 is expressed(owing to bilinearity) by

〈 gp(u) | v 〉 =

⟨gξ(q)

(uh ∂p

∂qh

∣∣∣q

)| vk ∂p

∂qk

∣∣∣q

⟩= uh

⟨gξ(q)

( ∂p

∂qh

∣∣∣q

)| ∂p

∂qk

∣∣∣q

⟩vk

and therefore gp is characterized by the (non-singular, symmetric) n×n matrixof components

ghk(q) :=

⟨gξ(q)

( ∂p

∂qh

∣∣∣q

)| ∂p

∂qk

∣∣∣q

⟩

Almost-symplectic structure

An almost-symplectic structure on a 2n-dimensional manifold M is a skewsymmetric vector bundle isomorphism of TM onto T ∗M , that is, a semi-basic 1-form

ω : TM → T ∗M : (π, X) 7→ (π, ωπ(X))

such that, for each π ∈ M , its restriction

ωπ : TπM → T ∗πM : X 7→ ωπ(X)

is a linear isomorphism (non-degenerateness) and the bilinear product 〈ωπ(X) |Y 〉 of any two vectors X,Y ∈ TπM satisfies the skew-symmetry condition(anticommutativity)

〈ωπ(X) | Y 〉 = −〈ωπ(Y ) | X〉

ω is said to be a symplectic structure, if it is characterized – in a suit-able atlas of charts of M – by the (non-singular, skew-symmetric) 2n × 2nsymplectic matrix of components[

ω(1)(1) ω(1)(2)

ω(2)(1) ω(2)(2)

]with ω(1)(1) = ω(2)(2) = 0 (zero n × n matrix) and ω(1)(2) = −ω(2)(1) = δ(identity n× n matrix).

On a cotangent bundle M = T ∗Q there exists a canonical symplecticstructure, characterized by the symplectic matrix in the atlas of the naturalcharts of M .

64 5 Appendix

5.5 Intrinsic approach to smooth manifods

A deeper insight into the structure of an embedded submanifold of a Eu-clidean affine space, will provide useful suggestions for a generalized definitionof ‘smooth manifold’ and an ‘intrinsic’ approach to its geometry (without anyreference to embeddings into Euclidean environments).

Intrinsic geometry of embedded submanifolds

Let Q be an n-dimensional, smooth manifold embedded into a Euclideanaffine space E .

The n-dimensional atlas A made up of all the charts of Q , could be shownto be

1. C∞, i.e. any two charts

ϕ = ξ−1 : U ⊂ Q → W ⊂ Rn , ϕ′ = ξ′−1 : U ′ ⊂ Q → W ′ ⊂ Rn

belonging to A are bijections (between subsets of Q and Rn , respectively)related to each other by C∞ transition functions

q ∈ ϕ(U ∩ U ′) ϕ−1

7−→ p ∈ U ∩ U ′ ϕ′7−→ q′ ∈ ϕ′(U ∩ U ′)

q′ ∈ ϕ′(U ∩ U ′) ϕ′−1

7−→ p ∈ U ∩ U ′ ϕ7−→ q ∈ ϕ(U ∩ U ′)

2. complete, i.e. any bijection

ϕ : U ⊂ Q → W ⊂ Rn

which is C∞-related to all the charts of A , belongs to A .

From the topological point of view, the domains of the charts belonging toa complete, C∞, n-dimensional atlas (on any set) turn out to be the base of a‘locally Euclidean’ manifold topology, where all the charts are homeomorphismsbetween open subsets (actually, the manifold topology determined by A onQ coincides with the subspace topology of Q ).

From the differential point of view, for each p ∈ Q , the tangent vectorspace TpQ can be identified – through a canonical isomorphism– with thequotient vector space Ap × Rn/ ∼p , where

Ap ⊂ A

5 Appendix 65

denotes the set of all the charts with domains containing p and

(ϕ, v) ∼p (ϕ′, v′) iff v′ = dϕ(p)(ϕ′ ◦ ϕ−1) (v) (?)

is the equivalence relation defined in Ap × Rn by the transformation law forthe components of tangent vectors. 20

The vector space structure of the quotient is the invariant one that makesan isomorphism of the bijection 21

ϕ′p : Rn → Ap × Rn/ ∼p : v 7−→ ϕ′p(v) := [(ϕ, v)]∼p (◦)

determined by any ϕ = ξ−1 ∈ Ap .

The above mentioned identification is due to the invariant isomorphism

ιp : TpQ → Ap × Rn/ ∼p : v = dϕ(p)ξ (v) 7−→ ιp(v) := ϕ′p(v)

In particular, for the basis of TpQ determined by a chart ϕ = ξ−1 ∈ Ap ,we put 22

∂p

∂qh

∣∣∣ϕ(p)

= dϕ(p)ξ (δh)ιp7−→ ∂

∂qh

∣∣∣p

:= ϕ′p(δh)

As a consequence, for the vector p(t) tangent to a smooth curve γ : t ∈I → γ(t) ∈ Q at p(t) = γ(t) , we have – in any chart ϕ = ξ−1 ∈ Aγ(t) (whereq(t) = ϕ(γ(t)) ) –

p(t) = dq(t)ξ (q(t))ιp7−→ γ(t) := ϕ′γ(t)(q(t))

= ϕ′γ(t)(qh(t) δh)

= qh(t) ϕ′γ(t)(δh)

= qh(t)∂

∂qh

∣∣∣γ(t)

20 For any two charts ϕ = ξ−1 and ϕ′ = ξ′−1 in Ap – where p = ξ(q) = ξ′(q′) – and anytangent vector v = dqξ (v) = dq′ξ′ (v′) ∈ TpQ , we have

dq′ξ′ (v′) = dqξ (v) = dq(ξ′ ◦ ϕ′ ◦ ϕ−1) (v) = dq′ξ′(dq(ϕ′ ◦ ϕ−1) (v)

)whence, owing to the injectivity of dq′ξ′ ,

v′ = dq(ϕ′ ◦ ϕ−1) (v)

21 [(ϕ, v)]∼p will denote the complete equivalence class of (ϕ, v) ∈ Ap × Rn under ∼p .22 (δh)h=1,...,n will denote the canonical basis of Rn .

66 5 Appendix

Intrinsic geometry of smooth manifolds

On a non-empty set M , a structure of m-dimensional smooth manifold willthen be defined as a complete, C∞, m-dimensional atlas A .

From the topological point of view, the manifold topology of M – i.e. thetopology determined by the domains of the charts belonging to A – is locallyEuclidean.

From the differential point of view, the smoothness of M – i.e. the differ-entiability of the transition functions of A – determines, at each π ∈ M , anm-dimensional tangent vector space given by

TπM := Aπ × Rm/ ∼π

(where the equivalence relation ∼π is defined as in (?) ).Each chart ϕ (or coordinate system x = (xi)i=1,...,m ) belonging to Aπ

determines an isomorphism ϕ′π : Rm → TπM (defined as in (◦) ) and then abasis in TπM given by

∂

∂xi

∣∣∣π

:= ϕ′π(δi)

Moreover, an invariant vector κ(t) tangent to a smooth curve κ : t ∈ I 7→κ(t) ∈ M at κ(t) is defined by putting – in any chart ϕ ∈ Aκ(t) (where thecoordinate expression x(t) = ϕ(κ(t)) is meant to be C∞ )

κ(t) := ϕ′κ(t)(x(t))

= ϕ′κ(t)(xi(t) δi)

= xi(t) ϕ′κ(t)(δi)

= xi(t)∂

∂xi

∣∣∣κ(t)

From the above definitions, all of the further developments of the theory(tangent bundles and differential equations, cotangent bundle and differentialforms) follow in quite a natural way.

67

References

[1] C. Godbillon, Geometrie Differentielle et MecaniqueAnalytique, Hermann,Paris (1969).

[2] R. Abraham and J.E. Marsden, Foundations of Mechanics, Benjamin/Cummings,Mass. (1978).

[3] P. Libermann and C. Marle, Symplectic Geometry and Analytical Mechan-ics, Reidel Publ., Dordrecht (1987).

[4] M. de Leon and P.R. Rodrigues, Methods of Differential Geometry in An-alytical Mechanics, North-Holland, Amsterdam (1989).

[5] M.J. Gotay, J.M. Nester and G. Hinds, Presymplectic Manifolds and theDirac-Bergmann Theory of Constraints, J. Math. Phys. 19 (1978), 2388-2399.

[6] M.J. Gotay and J.M. Nester Presymplectic Lagrangian Systems I and II,Ann. Inst. Henri poincare A 30 (1979), 129-142 and 32 (1980), 1-13.

[7] F. Cantrijn, J.F. Carinena, M. Crampin and L.A. Ibort, Reduction of De-generate Lagrangian systems, J. Geom. Phys. 3 (1986), 353-400.

[8] X. Gracia and J.M. Pons, A Generalized Geometric Framework for Con-strained Systems, Diff. Geom. Appl. 2 (1992) 223-247.

[9] M. de Leon and D. Martin de Diego, Symmetries and Constants of theMotion for Singular Lagrangian Systems, Int. J. Theor. Phys. 35 (1996),975-1011.

[10] G. Blankenstein and A.J. van der Schaft, Symmetry and Reduction inImplicit Generalized Hamiltonian Systems, Rep. Math. Phys. 47 (2001), 57-100.

[11] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry I,Wiley Intersc, Publ., New York (1963).

[12] W.M. Boothby, Introduction to Differentiable Manifolds and RiemannianGeometry, Academic Press, New York (1975).

[13] F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups,Springer-Verlag, New York (1983).

68

[14] J.M. Lee, Introduction to Smooth Manifolds, Springer-Verlag, New York(2003).

[15] P.J. Rabier and W.C. Rheinboldt, A Geometric Treatment of ImplicitDifferential-Algebraic Equations, J. Diff. Eqs 109 (1994),110-146.

[16] F. Barone and R. Grassini, Geometry of Implicit Differential Equationson Smooth Manifolds, Bull. Math.Soc. Sc. Math. Roumanie 47(95) No. 3-4(2004), 119-145.

[17] F. Barone and R. Grassini, Generalized Dirac Equations in Implicit Formfor Constrained Mechanical Systems, Ann. Henri Poincare 3 (2002), 297-330.

[18] F. Barone and R. Grassini, A Note on Godbillon Dynamics and GeneralizedPotentials, Ric. Mat. 51, 2 (2002), 265-284.

[19] F. Barone and R. Grassini, Generalized Hamiltonia Dynamics after Diracand Tulczyjew, Banach Center Publ. 59 (2003), 77-97.

[20] R. Grassini, Noether Symmetries and Conserved Momenta of Dirac Equa-tion in Presymplectic Dynamics, Int. Math. Forum, 2, no.45 (2007), 2207-2220.

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