Introduction to the. Geometry of Classical Dynamics

Transcription

1 Introduction to the Geometry of Classical Dynamics Renato Grassini Dipartimento di Matematica e Applicazioni Università di Napoli Federico II HIKARI LT D

2 HIKARI LTD Hikari Ltd is a publisher of international scientific journals and books. Renato Grassini, Introduction to the Geometry of Classical Dynamics, First published No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission of the publisher Hikari Ltd. ISBN Typeset using L A TEX. Mathematics Subject Classification: 70H45, 58F05, 34A09 Keywords: smooth manifolds, implicit differential equations, d Alembert s principle, Lagrangian and Hamiltonian dynamics Published by Hikari Ltd

3 iii Preface The aim of this paper is to lead (in most elementary terms) an undergraduate student of Mathematics or Physics from the historical Newtoniand 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-free language, apt to gain from the very historical formulation the consciousness (at an early stage) of the geometric structures that are intrinsic to the very nature of classical dynamics. The coordinate formalism will be confined to the ancillary role of providing simple proofs for some geometric results (which would otherwise require more advanced geometry), as well as re-obtaining the local analytical formulation of the theory from the global geometrical one. 2 The main conceptual tool of our approach will be the simple and general notion of differential equation in implicit form, which, treating an equation just as a subset extracted from the tangent bundle of some manifold through a geometric or algebraic property, will directly allow us to capture the structural core underlying the evolution law of classical dynamics. 3 1 Such an Introduction will cover the big gap existing in the current literature between the (empirical) elementary presentation of Newtonian-d Alembertian dynamics and the (abstract) differential-geometric formulation of Lagrangian-Hamiltonian dynamics. Standard textbooks 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 to smooth manifolds embedded in Euclidean affine spaces, and are listed in Appendix (whose reading is meant to preceed that of the main text). More advanced geometry can be found in 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] (on implicit differential equations) and [17][18][19][20] (on their role in advanced dynamics).

4 Contents Preface iii 1 From Newton to d Alembert The data Configuration space Mass distribution Force field Mechanical system The question Smooth motions Dynamically possible motions The answer after Newton Newton s law of constrained dynamics d Alembert s reformulation d Alembert s principle of virtual works d Alembert s implicit equation Tangent dynamically possible motions d Alembert equation From d Alembert to Lagrange Integrable part of d Alembert equation Restriction of d Alembert equation Extraction of the integrable part Lagrange equation Covector formulation Riemannian geodesic curvature field Normal form Integral curves Classical Lagrange equations iv

5 Table of Contents v 2.3 Euler-Lagrange equation Conservative system Lagrangian geodesic curvature field Integral curves Classical Euler-Lagrange equations Variational principle of stationary action Variational calculus Variational principle Riemannian case From Lagrange to Hamilton Legendre transformation Lagrangian function and Legendre transformation Coordinate formalism Hamiltonian function Energy and Hamiltonian function Coordinate formalism Hamiltonian vector field Canonical symplectic structure and Hamiltonian vector field Coordinate formalism Hamilton equation Cotangent dynamically possible motions Hamilton equation Classical Hamilton equations Concluding remarks Inertia and force Gauge transformations Geometrizing physical fields Geometrical dynamics Appendix Submanifolds of a Euclidean space Affine subspaces Embedded submanifolds Atlas of charts Locally Euclidean topology Smoothness Smooth curves and tangent vectors Tangent vector spaces Open submanifolds

6 vi Table of Contents Implicit function theorem Tangent bundle and differential equations Tangent bundle and canonical projection Vector fibre bundle structure Tangent lift Differential equations in implicit form Integral curves Integrable part Vector fields and normal form Cauchy problems and determinism Second tangent bundle and second-order differential equations. 52 Second tangent lift and second tangent bundle Affine fibre bundle structure Second-order differential equations in implicit form Integral curves and base integral curves Semi-sprays and normal form Cauchy problems and determinism Cotangent bundle and differential forms Cotangent vector spaces Cotangent bundle and canonical projection Vector bundle structure Differential 1-forms Semi-basic differential 1-forms Semi-Riemannian metric Almost-symplectic structure Intrinsic approach to smooth manifods Intrinsic geometry of embedded submanifolds Intrinsic geometry of smooth manifolds References 67

7 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 implicit form on a Euclidean space. 1.1 The data Classical particle dynamics deals with an empirical problem, whose data in the simplest cases can be described in mathematical terms as follows. Configuration space A reference space mathematical extension of a rigid body carrying an observer is conceived as a 3-dimensional, Euclidean, affine space E 3, modelled on a vector space E 3 with inner product (time will be conceived as an oriented, 1-dimensional, Euclidean, affine space, classically identified with the real line R ). Particle is synonymous with point-like body, i.e. a body whose position in E 3 is defined by a single point of E 3. Therefore, for a given (ordered) system of ν particles, a position or configuration in E 3 is defined by a single point of E := E 3 ν (3ν-dimensional, Euclidean, affine space, modelled on E := E 3 ν ). 1 1 Recall that the inner product in E is defined by putting, for all v = (v 1,..., v ν ) and w = (w 1,..., w ν ) belonging to E, v w := ν v i w i i=1 1

8 2 1 From Newton to d Alembert The particle system may generally be subject in E 3 to some holonomic (or positional) constraints, owing to which it is virtually allowed to occupy only the positions belonging to an embedded submanifold Q E called configuration space of the system in E 3. Q generally consists of all the points p E satisfying a number κ < 3ν of independent scalar equalities f α (p) = 0 (α = 1,..., κ), called two-sided constraints, and/or some strict scalar inequalities g β (p) > 0 (β = 1,..., µ), called strict one-sided constraints. Under usual hypotheses of regularity on the 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 is given by the dimension of the Euclidean environment minus the number of the two-sided constraints (in absence of two-sided constraints, Q is an open submanifold). 2 Mass distribution The response of the system to any internal or external influence, will generally depend on how massive its particles are, the inertial mass of a particle being conceived as a positive scalar quantity. The inertial mass distribution carried by the system will be denoted by m := (m 1,..., m ν ) Force field The force δύναµις resultant of all the internal and/or external influences acting in E 3 on the particles and not depending on their being constrained or unconstrained, is generally described as a smooth vector-valued mapping, called force field, F : U E T E E : (p, v) F(p, v) = (F 1 (p, v),..., F ν (p, v)) where U denotes an open subset of E containing Q. Remark that, if F {p} E = const. for all p U, then F called positional force field can be regarded as a smooth mapping f : U E E : p f(p), where f(p) denotes the constant value of F {p} E. 3 2 We shall not consider non-strict one-sided constraints, which would give rise to a manifold Q with boundary (nor shall we consider time-dependent constraints, which would give rise to a manifold Q fibred over the real line). 3 We shall not consider time-dependent force fields, which would later give rise to timedependent differential equations.

9 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 dynamics can be expressed in the following terms. Smooth motions A smooth motion of the particle system in the reference space E 3 as a smooth curve of E, say is described γ : I R E : t γ(t) = p(t) Such a curve γ is meant to establish a configuration p(t) at each instant t of a time interval I R, along which the positions p(t) = (p 1 (t),..., p ν (t)) of the particles, their velocities ṗ(t) = (ṗ 1 (t),..., ṗ ν (t)) and their accelerations p(t) = ( p 1 (t),..., p ν (t)) (as well as the derivatives of any order) vary continuously. Dynamically possible motions Smooth dynamics basically deals with the time-evolution problem in the unknown γ expressed by the following question: For the above constrained point-mass system, what are the smooth motions in the chosen reference space that are possible under the action of the given δύναµις? 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 ).

10 4 1 From Newton to d Alembert 1.3 The answer after Newton After Newton, the answer to the above predictive question is given by the following law. Newton s law of constrained dynamics A smooth curve γ : I E : t γ(t) = p(t) is a DPM of S, iff, for all t I, 4 p(t) Q m p(t) = F(p(t), ṗ(t)) + Φ(t) Φ(t) Tp(t)Q The first condition just exhibits the kinematical effects of the constraints, which only allow motions living in Q. The second condition is the classical Newton s law with a right hand side encompassing the possible dynamical effects of the contraints, expressed by an unknown constraint reaction Φ(t) E. The third condition expresses the only known empirical requisite of the constraint reaction, which apart from possible frictions tangent to Q is always orthogonal to Q d Alembert s reformulation After d Alembert, the unknown constraint reaction can be cancelled from Newton 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), ṗ(t)) m p(t) T p(t)q 4 For any µ = (µ 1,..., µ ν ) R ν and w = (w 1,..., w ν ) E, we shall put µ w := (µ 1 w 1,..., µ ν w n ). Moreover, for any p Q, T p Q will denote the orthogonal complement in E of the tangent vector space T p Q. 5 If the empirical law of friction is to be taken into consideration, then you will embody it in F.

11 1 From Newton to d Alembert 5 that is, ( ) F(p(t), ṗ(t)) m p(t) δp = 0, δp T p(t) Q called d Alembert s principle of virtual works (since the inner product therein defines the work of active force F(p(t), ṗ(t)) and inertial force m p(t) along any virtual displacement δp, i.e. any infinitesimal displacement tangent to Q and therefore virtually allowed by the constraints). 6 So a smooth curve γ : I E : t 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), ṗ(t)) δp, δp T p(t) Q ( ) 1.5 d Alembert s implicit equation From the mathematical point of view, condition ( ) shows that determining the DPM s of S is a second-order differential problem, whose unknown is a smooth curve of E. It turns into a first-order differential problem, whose unknown is a smooth curve of T E, as follows. Tangent dynamically possible motions If is DPM of S, its tangent lift γ : I E : t γ(t) = p(t) γ : I T E : t γ(t) = (p(t), ṗ(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 ( ) F i (p(t), ṗ(t)) m i p i (t) δp i = 0, δp = (δp 1,..., δp ν ) T p(t) Q Remark that, if Q is an open submanifold of E (absence of two-sided constraints), one has T p Q = E = {0} for all p Q (absence of constraint reaction) and then d Alembert s principle simply reads m i p i (t) = F i (p(t), ṗ(t)), i = 1,..., ν

12 6 1 From Newton to d Alembert DPM s and TDPM s bijectively correspond to one another, since the tangent lift operator γ γ is obviously inverted by the base projection operator γ γ. Trough such a bijection, the problem of determining the DPM s proves to be naturally equivalent to that of determining the TDPM s. Owing to ( ), a smooth curve c : I T E : t c(t) = (p(t), v(t)) is a TDPM of S, iff it satisfies, for all t I, the time-evolution law p(t) Q, ṗ(t) = v(t), m v(t) δp = F(p(t), v(t)) δp, δp T p(t) Q ( ) d Alembert equation Condition ( ) will now be seen to correspond to a first-order differential equation in implicit form on T E (second-order on E ), 7 namely d Alembert equation D d Al := {(p, v; u, w) T T E p Q, u = v, F(p, v) m w T p Q } = {(p, v; u, w) T T E p Q, u = v, m w δp = F(p, v) δp, δp T p Q } T 2 E Proposition 1 The TDPMs of S are the integral curves of D d Al the DPMs are its base integral curves). (and then Proof Recall that a smooth curve c : I T E : t c(t) = (p(t), v(t)) is an integral curve of D d Al, iff its tangent lift ċ : I T T E : t ċ(t) = (p(t), v(t); ṗ(t), v(t)) satisfies condition that is, for all t I, Im ċ D d Al ċ(t) = (p(t), v(t); ṗ(t), v(t)) D d Al which is exactly condition ( ), characterizing the TDPM s. 7 Recall that T E = E E is a Euclidean affine space modelled on E E. Its tangent bundle is therefore T T E = (E E) (E E).

13 1 From Newton to d Alembert 7 Also recall that a smooth curve γ : I T E : t γ(t) = p(t) is a base integral curve of D d Al, iff its second tangent lift γ : I T T E : t γ(t) = (p(t), ṗ(t); ṗ(t), p(t)) satisfies condition that is, for all t I, Im γ D d Al γ(t) = (p(t), ṗ(t); ṗ(t), p(t)) D d Al which is exactly condition ( ), characterizing the DPM s.

14 Chapter 2 From d Alembert to Lagrange Dynamics is now a problem of integration, i.e. determination and/or qualitative analysis of the integral curves of d Alembert equation (implicit differential equation on Euclidean space T E ). In this connection, the latter will be shown to be equivalent to a Lagrange equation (implicit differential equation on manifold T Q ), which will naturally be obtained and thoroughly discussed. 2.1 Integrable part of d Alembert equation As to the integration of D d Al, the first step is to extract its integrable part, i.e. the region D (i) d Al D d 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 D d Al are constrained to live in Q (see condition ( ) ) and then their tangent lifts, i.e. the integral curves, live in T Q. As a consequence, the tangent lifts of the integral curves live in T T Q, that is to say, D (i) d Al T T Q So we obtain D (i) d Al D d Al T T Q 8

15 2 From d Alembert to Lagrange 9 Restriction of d Alembert equation The above result suggests focusing on the restriction of D d Al obtained via intersection with T T Q, i.e. on the first-order differential equation in implicit form on T Q (second-order on Q ) D Lagr := D d Al T T Q which will be called Lagrange equation. Note that Lagrange equation is an effective restriction of d Alembert equation (i.e. D Lagr D d Al ) in presence, and only in presence, of two-sided constraints (when dim Q < dim E ), as is shown by the following proposition. Proposition 2 D Lagr D d Al, iff dim Q < dim E. Proof Let dim Q < dim E (whence T p Q E for all p Q ). We shall prove that, under the above hypothesis, D Lagr D d Al. Indeed, we can choose p Q, v E T p Q (whence (p, v) T Q ), u = v and w = 1 (F(p, v) + m Φ) with Φ Tp Q (whence F(p, v) mw Tp Q ) and clearly we obtain (p, v; u, w) D d Al, (p, v; u, w) T T Q and then (p, v; u, w) D Lagr. Conversely, let dim Q = dim E (whence T p Q = T(p,v) 2 Q = E for all (p, v) T Q ). We shall prove that, under the above hypothesis, D d Al D Lagr. Indeed, for any (p, v; u, w) D d Al, we have p Q, v E = T p Q, u = v and w E = T(p,v) 2 Q, that is, (p, v; u, w) T 2 Q T T Q and then (p, v; u, w) D Lagr. Extraction of the integrable part By extracting D Lagr from D d Al, via intersection of the latter with T T Q, we just obtain D (i) d Al, as is shown by the following proposition. Proposition 3 D (i) d Al = D Lagr Proof First we remark that D d Al and D Lagr are equivalent equations i.e. they have the same integral curves since, owing to inclusions D (i) d Al D Lagr D d Al, condition Im ċ D d Al (i.e. Im ċ D (i) d Al ) implies Im ċ D Lagr and, conversely, Im ċ D Lagr implies Im ċ D d Al. That amounts to saying D (i) d Al = D(i) Lagr. Then we anticipate that D Lagr is integrable, i.e. D Lagr = D (i) Lagr. Hence our claim.

16 10 2 From d Alembert to Lagrange So the focal point is now to prove the integrability of D Lagr. That will follow from the stronger property of D Lagr being reducible to normal form, as will be shown in the sequel. 2.2 Lagrange equation In order to prove the reducibility of Lagrange equation to normal form, we shall need to give a deeper insight into its algebraic formulation and the underlying geometric structures. Its integral curves will then be given a global characterization in terms of the above geometric structures (the traditional local characterization in coordinate formalism will finally be deduced). Covector formulation From the set-theoretical point of view, Lagrange equation can be expressed in the form D Lagr = {(p, v; u, w) T T Q u = v, m w δp = F(p, v) δp, δp T p Q } T 2 Q From the algebraic point of view, the condition on virtual works characterizing D Lagr can be given the form of a covector equality, as will now be shown. Associated with the inertial mass distribution m, there is a semi-basic 1-form [m] : T 2 Q T Q : (p, v; v, w) (p, [m](p, v, w)) on T 2 Q, called covector inertial field, whose value [m](p, v, w) := (m w) TpQ T p Q at any (p, v; v, w) T 2 Q is the virtual work of m w. 1 Associated with the force field F, there is a semi-basic 1-form F : T Q T Q : (p, v) (p, F (p, v)) on T Q, called covector force field, whose value F (p, v) := F(p, v) TpQ T p Q at any (p, v) T Q is the virtual work of F(p, v). 2 1 For any u E, we define u E by putting u : v E u v R. Then, for any p Q, the restriction of u to T p Q yields u TpQ T p Q. 2 The virtual work of a positional force field f can be regarded as an ordinary 1-form on Q, namely f : p Q (p, f(p)) T Q, f(p) := f(p) TpQ T p Q.

17 2 From d Alembert to Lagrange 11 Proposition 4 Lagrange equation can be given the covector formulation D Lagr = {(p, v; u, w) T T Q u = v, [m](p, v, w) = F (p, v)} Proof Just notice that condition m w δp = F(p, v) δp, δp T p Q means (m w) TpQ = F(p, v) TpQ that is, [m](p, v, w) = F (p, v) Riemannian geodesic curvature field The reducibility of D Lagr to normal form requires that, for any choice of the data (p, v) T Q, the algebraic equation [m](p, v, w) = F (p, v) should be uniquely solvable with respect to the unknown w T(p,v) 2 Q. As the latter only appears in the left hand side of the equation, the above property is to be checked through a thorough investigation of [m]. The following geometric considerations showing that such a semi-basic 1-form on T 2 Q is the transformed of a suitable semi-basic 1-form on T Q through a distinguished semi-spray will prove to be crucial. Remark that [m] is the semi-basic 1-form induced on T 2 Q by the Euclidean metric g m : E E : u g m (u) := (m u) (positive definite, symmetric, linear map), since its value at any (p, v; v, w) T 2 Q is [m](p, v, w) = g m (w) Tp Q TpQ In the same way, there is a semi-basic 1-form g : T Q T Q : (p, v) (p, g p (v)) induced on T Q by g m, whose value at any (p, v) T Q is g p (v) := g m (v) Tp Q TpQ

18 12 2 From d Alembert to Lagrange g is a Riemannian metric on Q, characterized by the quadratic form or Lagrangian function K : T Q R : (p, v) K(p, v) := 1 2 g p(v) v = 1 2 m v v (the kinetic energy of the mechanical system). Riemannian manifold (Q, K) carries a distinguished semi-spray Γ K : T Q T 2 Q : (p, v) (p, v; v, Γ K (p, v)) called Riemannian spray, uniquely determined by the following property. Proposition 5 There exists one, and only one, semi-spray Γ K for all (p, v) T Q, g m (Γ K (p, v)) = 0 TpQ on T Q s.t., Proof (i) Unicity First remark that, for any (p, v) T Q, the map w T(p,v)Q 2 (α) g m (w) Tp Q TpQ is injective, since g m (w 1 ) = g m (w 2 ) TpQ TpQ with w 1, w 2 T 2 (p,v) Q and then w 1 w 2 T p Q implies g p (w 1 w 2 ) = g m (w 1 w 2 ) TpQ = g m (w 1 ) g m (w 2 ) TpQ = 0 TpQ that is, recalling that g p : T p Q T p Q is a linear isomorphism, w 1 = w 2 So, if there exists a vector in T(p,v) 2 Q whose image through (α) is zero, it is unique. (ii) Existence For any (p, v) T Q, put Γ K (p, v) := w + u T 2 (p,v)q

19 2 From d Alembert to Lagrange 13 with w T 2 (p,v)q, ( u := gp 1 g m (w) TpQ ) T p Q The image of Γ K (p, v) through (α) is zero, since g m (Γ K (p, v)) = g m (w + u) TpQ TpQ = g m (w) + g m (u) TpQ = g m (w) + g p (u) TpQ = g m (w) g m (w) TpQ Γ K = 0 transforms g (semi-basic 1-form on T Q ) into TpQ TpQ [K] : T 2 Q T Q : (p, v; v, w) (p, [K](p, v, w)) (semi-basic 1-form on T 2 Q ) by putting, for any (p, v; v, w) T 2 Q, [K](p, v, w) := g p (w Γ K (p, v)) T p Q [K] will be called Riemannian geodesic curvature field. Actually [K] does not differ from [m], as is shown in the following proposition. Proposition 6 Lagrange equation, in covector formulation, reads Proof D Lagr = {(p, v; u, w) T T Q u = v, [K](p, v, w) = F (p, v)} Owing to Prop. 4, it will suffice to show that [K] = [m] To this end, note that, for any (p, v; v, w) T 2 Q, from Prop. 5 we obtain [K](p, v, w) := g p (w Γ K (p, v)) = g m (w Γ K (p, v)) TpQ = g m (w) g m (Γ K (p, v)) TpQ = g m (w) TpQ = [m](p, v, w) TpQ

20 14 2 From d Alembert to Lagrange Normal form The reducibility of Lagrange equation to normal form immediately follows from Prop. 6. Consider the vertical force field and the semi-spray F := g 1 F : T Q T Q : (p, v) (p, F (p, v)) F (p, v) := gp 1 (F (p, v)) T p Q Γ := Γ K + F : T Q T 2 Q : (p, v) (p, v; v, Γ(p, v)) Γ(p, v) := Γ K (p, v) + F (p, v) T 2 (p,v)q Proposition 7 Lagrange equation can be put in the normal form D Lagr = Im Γ = {(p, v; u, w) T T Q u = v, w = Γ(p, v)} Proof reads Just notice that covector equality [K](p, v, w) = F (p, v) g p (w Γ K (p, v)) = F (p, v) w Γ K (p, v) = gp 1 (F (p, v)) w = Γ K (p, v)) + F (p, v) w = Γ(p, v) Integral curves The condition characterizing the integral curves of Lagrange equation can now be formulated as follows. Let be a smooth curve of T Q and its projection onto Q. 3 c : I T Q : t c(t) = (p(t), v(t)) τ Q c : I Q : t (τ Q c)(t) = p(t) 3 The tangent lift of τ Q c will be denoted by (τ Q c).

21 2 From d Alembert to Lagrange 15 Proposition 8 c is an integral curve of D Lagr, iff (τ Q c) = c, [K] ċ = F c ( ) or, in normal form, ċ = Γ c ( ) Proof As is known, c is an integral curve of D Lagr, iff Im ċ D Lagr that is to say, for all t I, ( ) ċ(t) = (p(t), v(t); ṗ(t), v(t)) D Lagr (i) Owing to Prop. 6, condition ( ) reads ṗ(t) = v(t), [K](p(t), v(t), v(t)) = F (p(t), v(t)) that is, (τ Q c) (t) = (p(t), ṗ(t)) = (p(t), v(t)) = c(t) and ([K] ċ)(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 ṗ(t) = v(t), v(t) = Γ(p(t), v(t)) that is, ċ(t) = (p(t), v(t); ṗ(t), v(t)) = (p(t), v(t); v(t), Γ(p(t), v(t))) = (Γ c)(t) That proves our second claim.

22 16 2 From d Alembert to Lagrange As a consequence, the condition characterizing the base integral curves of Lagrange equation will be formulated as follows. Let be a smooth curve of Q and γ : I Q : t γ(t) = p(t) [K] γ : I T Q its Riemannian geodesic curvature, whose vector (rather than covector) expression is the covariant derivative 4 γ dt := g 1 [K] γ : I T Q Proposition 9 γ is a base integral curve of D Lagr, iff [K] γ = F γ ( ) or, in vector formulation, or, in normal form, γ dt = F γ γ = Γ γ ( ) Proof Recall that a base integral curve of Lagrange equation is the projection γ = τ Q c of an integral curve c (smooth curve of T Q satisfying condition ( ) or the equivalent ( )). (i) Now, if γ is a base integral curve, condition ( ) implies γ = (τ Q c) = c whence γ = ċ and then ( ). Conversely, if γ satisfies condition ( ), then it is 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).

23 2 From d Alembert to Lagrange 17 (ii) In the same way, through condition ( ), one can show that the base integral curves are characterized by ( ). Alternatively, from γ dt : t I γ [K] g 1 (p(t), ṗ(t); ṗ(t), p(t)) T 2 Q ( ) p(t), g p(t) ( p(t) Γ K (p(t), ṗ(t))) T Q ( ) p(t), p(t) Γ K (p(t), ṗ(t)) T Q and that is, γ Γ K γ : t I (p(t), p(t) Γ K (p(t).ṗ(t))) T Q γ dt = γ Γ K γ we deduce that the vector formulation of ( ) (which has already been seen to characterize the base integral curves) is equivalent to γ = Γ K 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 Riemannian geodesic curvature [K] γ = 0 coincide with the inertial motions of S (i.e. the motions which would be possible if F were zero). The effect of a covector force field F 0 is then that of deviating the DPM s of S from the inertial trend, by giving them a non-vanishing Riemannian geodesic curvature, namely [K] γ = F γ. Classical Lagrange equations The scalar equations obtained with the aid of a chart by orderly equalling the components of the covector or vector-valued functions which appear in the left and right hand sides of ( ) or ( ), are the classical Lagrange equations of Analytical Dynamics. They will prove to be only locally equivalent to the geometric Lagrange equation, in the sense that they only characterize the base integral curves of the latter which live in the coordinate domain of the given chart.

24 18 2 From d Alembert to Lagrange Preliminaries Consider the coordinate domain U = Im ξ of a chart ξ : q W ξ(q) U, expressing the points p U Q in function of coordinates q W R n (with n := dim Q ). 5 To any smooth curve γ : t I γ(t) = p(t) Q living in U, i.e. satisfying p = p(t) U for 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 (dependence from time t is understood). with To the first tangent lift c = γ, that is, p = ξ(q) (1) p = p(t) U, v = v(t) T p(t) Q v = ṗ there corresponds in ξ a smooth coordinate expression q = q(t) W, v = v(t) R n related to γ by (1) and the time derivative of (1) 6 where v = v(q, v) := v h p q h q (2) v = q is the n-tuple of linear components of v in ξ. Remark that, for all h = 1,..., n, v (q,v) = p q v (q,v), = v k 2 p q = d p q (3) v h q h q h q h q k dt q h ( ) 5 Recall that, for any q = ξ 1 p q (p) W, the partial derivatives q,..., p q 1 q n provide a basis of T p Q. 6 A repeated index, in upper and lower position, denotes summation over (1,..., n).

25 2 From d Alembert to Lagrange 19 with To the second tangent lift ċ = γ, that is, p = p(t) U, v = v(t) T p(t) Q, w = w(t) T 2 (p(t),v(t))q v = ṗ, w = v = p there corresponds in ξ a smooth coordinate expression q = q(t) W, v = v(t) R n, w = w(t) R n related to γ by (1), (2) and the time derivative of (2) where w = w(q, v, w) := w h p q + v h v k 2 p q q h q h q k 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 F h (q, v) := (F γ) h = (F (p, v)) h = F (p, v) p q = F(p, v) p q q h q h 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 1 p (F (p, v))) i = g ih (q) (F (p, v)) h = g ih (q) F h (q, v) where [g hk (q)] is the inverse of the nonsingular matrix [g hk (q)] defined by g hk (q) := g p ( p q h q ) p q = m p q p q q k q h q k Lagrange equations The above coordinate formalism will now be adopted for the characterization of the base integral curves of D Lagr living in the given coordinate domain.

26 20 2 From d Alembert to Lagrange Proposition 10 A smooth curve γ : t I p = ξ(q(t)) U Q living in the coordinate domain U of a chart ξ of Q is a base integral curve of D Lagr, iff its coordinate expression q = q(t) satisfies (for all h, i = 1,..., n ) the classical Lagrange equations q = v, or, in normal form, 7 q = v, d dt K v h (q,v) K q h (q,v) = F h (q, v) { } i v i = v j v k + F i (q, v) jk q ( ) h ( ) i Proof (i) Recall that γ is a base integral curve of D Lagr, iff it satisfies equation ( ). As γ lives in the coordinate domain of ξ, equation ( ) is equivalent to the n scalar equations obtained by orderly equalling the components in ξ of its left and right hand sides, i.e. ([K] γ) h = (F γ) h ( ) h The components F h (q, v) := (F γ) h have already been shown in the above preliminaries, where we have put v = q. The components ([K] γ) h will now be 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 q q h = mw(q, v, w) p q q h = d ( mv(q, v) p ) q mv(q, v) d p q dt q h dt q h = d ( mv(q, v) v ) (q,v) mv(q, v) v (q,v) dt v h q h 7 Here we encounter the Christhoffel symbols (of the Levi-Civita connection) associated with a Riemannian manifold, defined on the coordinate domain of any chart by { } i := 1 jk 2 gih ( j g kh + k g hj h g jk ) with i g hk := g hk q i

27 2 From d Alembert to Lagrange 21 ( ) 1 2 mv v = d (q,v) dt v h = d K (q,v) K (q,v) dt v h q h So equations ( ) h just take the form ( ) h. ( ) (q,v) 1 q h 2 mv v (ii) Also recall that γ is a base integral curve of D Lagr, iff it satisfies equation ( ). As γ lives in the coordinate domain of ξ, equation ( ) is equivalent to the n scalar equations obtained by orderly equalling the affine components in ξ of its left and right hand sides, i.e. γ i = (Γ γ) i ( ) i where γ i = w i = v i and (Γ γ) i = (Γ K γ + F γ) i = (Γ K γ) i + ( F γ) i = Γ i K(q, v) + F i (q, v) The components F i (q, v) := ( F γ) i have already been shown in the above preliminaries. The components Γ i K (q, v) := (Γ K γ) i will now be evaluated. To that purpose we need the coordinate expression of K, which is given by K(p, v) = 1 ( g p v h p ) q v k p q 2 q h q k = 1 2 g hk(q) v h v k whence and K (q,v) q h = 1 2 ( hg jk ) q v j v k K (q,v) v h = g hk (q)v k 2 K (q,v) v h q k = ( k g hj ) q v j 2 K (q,v) v h v k = g hk (q)

28 22 2 From d Alembert to Lagrange As a consequence, the above components ([K] γ) h expressed in the form can more explicitly be ([K] γ) h = ([K](p, v, w)) h = 2 K (q,v) w k + 2 K (q,v) v k K (q,v) v h v k v h q k q h whence = g hk (q)w k + ( k g hj ) q v j v k 1 2 ( hg jk ) q v j v k = g hk (q)w k ( kg hj ) q v j v k ( jg hk ) q v j v k 1 2 ( hg jk ) q v j v k = g hk (q)w k ( kg hj + j g kh h g jk ) q v j v k ( ) i γ = (g 1 ([K] γ)) i dt = (gp 1 ([K](p, v, w))) i = g ih (q) ([K](p, v, w)) { } h i = w i + v j v k jk q Therefore, identities 8 ( read Hence we obtain w i + ) i γ dt = ( γ Γ K γ) i = γ i (Γ K γ) i { } i v j v k = w i Γ i jk K(q, v) q { } i Γ i K(q, v) = v j v k jk q So equations ( ) i just take the form ( ) i. 8 See the proof, part (ii), of Prop. 9.

29 2 From d Alembert to Lagrange Euler-Lagrange equation A special mention, for its primary role in both mathematical and theoretical physics, 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 : T Q R : (p, v) E(p, v) := K(p, v) + V (p) (kinetic energy plus potential energy). Proposition 11 Along each integral curve c of D Lagr = {(p, v; u, w) T T Q u = v, [K](p, v, w) = d p V } the mechanical energy E keeps constant, i.e. E c = const. Proof If c = (p, v) denotes an integral curve and w := v, from ṗ = v and [K](p, v, w) = d p V it follows that d dt (E c) = d E(p, v) dt = d dt K(p, v) + d dt V (p) = d dt (1 2 m v v) + d dt V (p) Hence our claim. = m w v + d dt V (p) = [K](p, v, w) v + d p V v = d p V v + d p V v = 0

30 24 2 From d Alembert to Lagrange Lagrangian geodesic curvature field In conservative dynamics, the kinetic energy and the potential energy which are the two ingredients generating D Lagr can be merged into a unique object, as follows. Define a new Lagrangian function by putting L := K V : T Q R : (p, v) L(p, v) := K(p, v) V (p) (kinetic energy minus potential energy). Associated with L, there is a Lagrangian geodesic curvature field given by with [L] := [K] + dv : T 2 Q T Q : (p, v; v, w) (p, [L](p, v, w)) [L](p, v, w) := [K](p, v, w) + d p V T p Q (Riemannian geodesic curvature field minus conservative field). From the above definitions, it follows that that is, D Lagr D Lagr = D Eul Lagr does not differ from the Euler-Lagrange equation D Eul Lagr := {(p, v; u, w) T T Q u = v, [L](p, v, w) = 0} generated by L. Integral curves The conditions characterizing the integral curves and the base integral curves of Euler-Lagrange equation can now be formulated as follows. Proposition 12 A smooth curve c of T Q is an integral curve of D Eul Lagr, iff (τ Q c) = c, [L] ċ = 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 D Eul Lagr, iff its Lagrangian geodesic curvature vanishes, i.e. [L] γ = 0 ( ) Proof The same as the proof, part (i), of Prop.9.

31 2 From d Alembert to Lagrange 25 Classical Euler-Lagrange equations The local scalar equations obtained with the aid of a chart by equalling to zero the components of the covector-valued function which appears in the left hand side of ( ), are the classical Euler-Lagrange equations of Analytical Dynamics. Proposition 14 A smooth curve γ : t I p = ξ(q(t)) U Q living in the coordinate domain U of a chart ξ of Q is a base integral curve of D Eul Lagr, iff its coordinate expression q = q(t) satisfies (for all h = 1,..., n ) the classical Euler-Lagrange equations q = v, d L (q,v) L (q,v) = 0 dt v h q h ( ) h Proof Recall that γ is a base integral curve of D Eul Lagr, iff it satisfies equation ( ). As γ lives in the coordinate domain of ξ, equation ( ) is equivalent to the n scalar equations obtained by equalling to zero the components in ξ of 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) = [K](p, v, w) where we have put v = q. p q q h p q h q + = d K (q,v) K (q,v) + V q dt v h q h q h = d (K V ) (q,v) (K V ) dt v h q h = d L (q,v) L (q,v) dt v h q h So equations ( ) h just take the form ( ) h. d p V (q,v) p q q h

32 26 2 From d Alembert to Lagrange Remark For both L := K and L := K V, the semi-basic 1-form [L] : T 2 Q T Q is a mapping which takes any (p, v; v, w) T 2 Q to the covector (p, [L](p, v, w)) T Q characterized, in a chart, by components ([L](p, v, w)) h = 2 L (q,v) w k + 2 L (q,v) v k L (q,v) v h v k v h q k q h Such a coordinate technique can be used to define [L] for an arbitrary Lagrangian function L : S T Q R (smooth on an open subset S of T Q ), since also in such a case the value [L](p, v, w) (for any (p, v) S ), defined by the above components ([L](p, v, w)) h in a given chart, turns out to be an invariant covector 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 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 (with ɛ > 0 and J I ) satisfying χ : ( ɛ, ɛ) J Q : (s, t) p(s, t) p(0, t) = p(t), t J and, at the end-points of a closed interval [t 1, t 2 ] J, p(s, t 1 ) = p(t 1 ), p(s, t 2 ) = p(t 2 ), s ( ɛ, ɛ) χ can be thought of as a one-parameter family { χ s : J Q : t p s (t) := p(s, t) } s ( ɛ,ɛ) 9 Actually, through higher geometric methods, [L] can be given a coordinate-free definition.

33 2 From d Alembert to Lagrange 27 of varied curves near γ J = χ o with fixed end-points in [t 1, t 2 ] J I, whose tangent lifts { χ s } s ( ɛ,ɛ) are all included in ( χ : ( ɛ, ɛ) J T Q : (s, t) p(s, t), p ) t (s,t) χ also define a one-parameter family { χ t : ( ɛ, ɛ) Q : s p t (s) := p(s, t) } t J of isocronous curves, whose tangent vectors at the points of γ J are the values of ( χ o : J T Q : t p(t), p ) s (0,t) Now consider, in a closed sub-interval [t 1, t 2 ] of I, the action of γ, i.e. the integral 10 t2 Iγ := (L γ) dt t 1 For any smooth variation χ of γ with fixed end-points in [t 1, t 2 ], the action of χ is then the real-valued function I χ := t2 t 1 (L χ) dt : ( ɛ, ɛ) R : s I χs := At s = 0, the value of I χ is I γ and its derivative δ γ I χ := di χ ds is called the first variation of I χ at γ. 0 t2 t 1 (L χ s ) dt Proposition 15 For every smooth variation χ of γ with fixed end-points in a closed sub-interval [t 1, t 2 ] of I, the first variation of I χ at γ is related to the Lagrangian geodesic curvature of γ by t2 δ γ I χ = [L] γ χ o dt t 1 10 All of the following considerations and results of variational theory, hold true for an arbitrary smooth Lagrangian function as well (on this matter, also recall the final Remark of the previous section).

34 28 2 From d Alembert to Lagrange Proof First remark that δ γ I χ := d t2 (L χ) dt = ds 0 t 1 t2 t 1 (L χ) s dt s=0 Now, for any given t o J, consider a chart ξ on a neighbourhood U of the point p(t o ) Im γ. In a suitably small neighbourhood of (0, t o ) ( ɛ, ɛ) J, χ takes values in U (by continuity) and will then have a local coordinate expression (q h (s, t)). As a consequence, χ will have local coordinate expression ( q h (s, t), qh t and χ o (L χ) s ), denoted, for s = 0, by (q = q(t), v = v(t)) with v = q, (s,t) will have components χ o h := qh s. With the aid of ξ, we obtain s=0 that is, 11 (0,to) = L q h + q (q(to),v(t L 2 q h h o)) s (0,to) v (q(to),v(t h o)) s t (0,to) = L q h + q (q(to),v(t h o)) s (0,to) d ( L (q,v) q h ) ( d L ) (q,v) q h dt to v h s s=0 dt to v h s (0,to) ( L = q (q(to),v(t d L ) (q,v) χ h o)) dt to v h o h (t o ) + d ( L ) (q,v) χ dt to v h o h (L χ) s (0,to) = [L] γ χ o (t o ) + d F L γ χ dt to o So, on the whole J, we have (L χ) = [L] γ χ s s=0 o + d dt F L γ χ o As χ o(t 1 ) = 0 and χ o(t 2 ) = 0, we also have t2 t 1 Hence our claim. d dt F L γ χ o dt = F L γ χ o (t 2 ) F L γ χ o (t 1 ) = 0 11 F L γ : I T Q will denote the (invariant) covector-valued map whose components in ξ are given by (F L γ) h = L v h (q,v) (see the final Remark of Legendre transformation in Chap. 3).

35 2 From d Alembert to Lagrange 29 Variational principle A smooth curve γ : I Q is said to be a geodesic curve of (Q, L), if it satisfies Hamilton s variational principle of stationary action, owing to which, for every smooth variation χ of γ with fixed end-points in a closed sub-interval of I, I χ is required to be stationary at γ, that is to say, δ γ I χ = 0, χ The above variational principle is completely equivalent to Euler-Lagrange equation. Proposition 16 The geodesic curves of (Q, L) are the base integral curves of D Eul Lagr. Proof (i) If γ : I Q is a base integral curve of D Eul Lagr, i.e. [L] γ = 0 we clearly obtain, for every smooth variation χ of γ with fixed end-points in a closed sub-interval [t 1, t 2 ] of I, t2 t 1 [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 p(t) Q is not a base integral curve of D Eul Lagr, say ([L] γ)(t o ) 0, t o I we shall prove that, for a suitable smooth variation χ of γ with fixed endpoints in a closed sub-interval [t 1, t 2 ] of I, t2 t 1 [L] γ χ o dt 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 p(t o ) Owing to our hypothesis, at least one of the components ([L] γ) h at t o, say ([L] γ) 1 (t o ) > 0 is non-null

36 30 2 From d Alembert to Lagrange By continuity, there exists a suitably small open interval J I containing t o s.t., for all t J, p(t) U, ([L] γ) 1 (t) > 0 Now, for each s ( ɛ, ɛ) with a suitably small ɛ > 0 and each t J, we consider the point p(s, t) U of coordinates q 1 (s, t) q 2 (s, t) q n (s, t) := q 1 (t) + s(cos(t t o ) cosα) := q 2 (t) := q n (t) where (q h (t)) is the n-tuple of coordinates of p(t) and 0 < α < π 2 By doing so, we define a map s.t. [t o α, t o + α] J χ : ( ɛ, ɛ) J Q : (s, t) p(s, t) which is immediately seen to be a smooth variation of γ with fixed end-points in [t 1, t 2 ] := [t o α, t o + α] The components of χ o are, for all t J, χ o 1 (t) χ o 2 (t) = χ o n (t) = 0 = cos(t t o ) cosα and, for all t (t 1, t 2 ), χ o 1 (t) > 0 Hence t2 t 1 [L] γ χ o dt = t2 t 1 ([L] γ) h χ o h dt = t2 t 1 ([L] γ) 1 χ o 1 dt > 0 which is our claim.

37 2 From d Alembert to Lagrange 31 Riemannian case For L = K, the above variational theory characterizes the geodesic curves of Riemannian manifold (Q, K) through condition [K] γ = 0 Owing to the positive definiteness of K, from the conservation law of energy 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 geodesic curve is a uniform motion, that is, K γ = const. > 0 The geometric meaning of non-degenerate geodesic curves will now be shown. Let us consider one more Lagrangian function, namely Λ := 2K which is clearly smooth on T Q \ K 1 (0). 12 If γ : t I Q : t p(t) Q is a smooth curve satisfying Im γ T Q \ K 1 (0) (i.e. ṗ(t) 0 for all t I ) and [t 1, t 2 ] I, the integral L γ := t2 t 1 (Λ γ) dt defines the length of the arc of γ with end-points (p(t 1 ), p(t 2 )) Λ is smooth only on T Q \ K 1 (0), since, in any chart, its partial derivatives Λ (q,v) 1 K (q,v) q h = 2K(q, v) q h, Λ (q,v) 1 K (q,v) v h = 2K(q, v) v h 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), ṗ(t)) dt = g p(t) (dp) dp > 0 defines the length, in the given Riemannian metric, of the infinitesimal arc dp := ṗ(t)dt.

38 32 2 From d Alembert to Lagrange The variational theory previously sketched, applied to Λ, 14 shows that γ is a curve of stationary length, that is, δ γ L χ = 0, χ iff [Λ] γ = 0 Proposition 17 γ is a non-degenerate geodesic curve of (Q, K), iff it is a uniform motion of stationary length. Proof The above result is an immediate consequence of the following Lemma 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 p(t) Q, we have (for any given t I and any chart at p(t) ) 15 ([Λ] γ) h = d Λ (q,v) Λ (q,v) dt v h q h = d ( 1 K ) (q,v) 1 K (q,v) dt 2κ v h 2κ q ( h 1 d K (q,v) = K ) (q,v) 2κ dt v h q h 1 = ([K] γ) h 2κ which is our claim. 14 See footnote See footnote 12.

39 Chapter 3 From Lagrange to Hamilton The Lagrangian dynamics of a conservative system S whose geometrical arena is the velocity phase space T Q, supporting Euler-Lagrange equation will now be taken into the range of Hamiltonian dynamics, where the geometrical arena is the momentum phase space T Q, supporting a Hamilton equation 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 provided by the well known map which takes any velocity (p, v) T Q onto the corresponding momentum (p, m v TpQ) T Q. Lagrangian function and Legendre transformation The above map is indeed the vector bundle isomorphism g : T Q T Q : (p, v) (p, g p (v)), g p (v) := m v TpQ owing to which the configuration space of the system has been given the structure of a Riemannian manifold (Q, K). 1 Such an isomorphism is also said to be the Legendre transformation determined by Lagrangian function L = K V (in terms of which it can be expressed) 2 and is denoted by F L : T Q T Q : (p, v) (p, F p L(v)), (whence π Q F L = τ Q ). 1 See Riemannian geodesic curvature field in Chap See the next Coordinate formalism. F p L(v) := g p (v) 33

40 34 3 From Lagrange to Hamilton Coordinate formalism Recall that T Q is a 2n-dimensional manifold, where each element (p, π) is completely characterized, in a natural chart, by two n-tuples of coordinates (q, p), namely the coordinates q = (q 1,..., q n ) of p = ξ(q), in a chart ξ of Q, and the components p = (p 1,..., p n ) of π in ξ, given by p h = π p q h q. In natural coordinate formalism, Legendre transformation (p, v) (p, π), π = F p L(v) is expressed by 3 (q, v) (q, p), p h = g hk (q) v k = L v h (q,v) (1) The inverse transformation (p, π) (p, v), v = (F p L) 1 (π) is in turn expressed by (q, p) (q, v), v h = g hk (q) p k =: ν h (q, p) (2) Remark Such a coordinate technique can be used to define a Legendre morphism F L for an arbitrary Lagrangian function L : S T Q T Q (smooth on an open subset S of T Q ), since also in such case the value π = F p L(v) Tp Q (for any (p, v) S ), defined by the above components p h = L v h (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, according to whether F L is injective or not. 3 Recall that p h = g p (v) p q ( p q ) ( p q ) p q = gp v = gξ(q) v k q h q h q h q k = g hk (q) v k = K (q,v) v h = L (q,v) v h 4 Actually, through higher geometric methods, F L can be given a coordinate-free definition.

41 3 From Lagrange to Hamilton Hamiltonian function Legendre transformation also determines transition from the Lagrangian function (on T Q ) to a Hamiltonian function (on T Q ). Energy and Hamiltonian function Recall that, associated with Lagrangian function L = K V, there is the energy function 5 E := K + V = 2K L = F L id TQ L : T Q R The Hamiltonian function H := E (F L) 1 : T Q R is the push forward of E by Legendre transformation F L. Coordinate formalism In natural coordinate formalism, the Hamiltonian function (p, π) H(p, π) = E((F L) 1 (p, π)) = E(p, (F p L) 1 (π)) = E(p, v) = F p L(v) v L(p, v) is expressed by = π v L(p, v), v = (F p L) 1 (π) (q, p) H(q, p) = p k v k L(q, v), v = ν(q, p) Hence, owing to (1) and (2), we obtain H q h (q,p) = p k v k q h (q,p) L v k (q,v) v k q h (q,p) L q h (q,v) = L q h (q,v) (3) and H p h (q,p) = v h + p k v k p h (q,p) L v k (q,v) v k p h (q,p) = ν h (q, p) (4) 5 We put F L id TQ : (p, v) T Q F p L(v) id TQ (v) = F p L(v) v = g p (v) v = 2K(p, v) R