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A Crash Course in Geometric Mechanics Tudos Ratiu

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A Crash Course in Geometric Mechanics Tudos Ratiu A Crash Course in Geometric Mechanics Tudos Ratiu To cite this version: Tudos Ratiu. A Crash Course in Geometric Mechanics. 3rd cycle. Monastir (Tunisie), 2005, pp.134. cel-00391890 HAL Id: cel-00391890 https://cel.archives-ouvertes.fr/cel-00391890 Submitted on 5 Jun 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. V A Crash Course in Geometric Mechanics T.S. Ratiu, R. Tudoran, L. Sbano, E. Sousa Dias & G. Terra Notes of the courses given by Tudor Ratiu 1 Introduction These lecture notes are the direct result of presentations held in two consec- utive years (2000 and 2001) at the Peyresq Conference Center, in the Alpes de Haute Provence, North of Nice. These summer schools were organized in conjunction with the Research Training Network MASIE (Mechanics and Symmetry in Europe) of the Fifth Framework Program of the European Com- mission and were intended for graduate students and postdocs who needed a crash course in geometric mechanics. They were also tailored to link with the other lectures and provide the necessary background for them. There are already many books on this subject and its links to symplectic and Poisson geometry (see, e.g. [AbMa78], [Arnold79], [GuSt84], [JoSa98], [LiMa87], [MaRa94], [McDSal95]) and the literature on this subject is overwhelming. So the goal of these two one-week intensive lectures was to find a quick way through this subject and give the young researchers enough tools to be able to sift and sort through the books and papers necessary for their own work. This is why these lectures present occasionally detailed proofs and sometimes only quick surveys of more extensive subjects that are, however, explained with care. The examples, on the other hand, are all carried out with detailed com- putations in order to show how one applies the theory in concrete cases. There are, essentially, four main examples that reappear throughout the lectures: par- ticle dynamics, the free and heavy tops, the motion of a charged particle in a magnetic field, and ideal incompressible fluid flow as well as related systems such as the Korteweg-de Vries and the Camassa-Holm equations. Each exam- ple illustrates several different constructions prevalent in geometric mechanics and is at the root of many developments and generalizations. 1 2 VACrash Course in Geometric Mechanics There is nothing original in these lectures and they are entirely based on three main sources: [MaRa94], the yet unfinished book [MaRa03], and some unpublished notes [MaRa95] on the geometric theory of fluid dynamics. When carrying out reduction, several strong regularity hypotheses will be made. The singular case is considerably more involved and we refer to [OR04] for an in- depth analysis of this case. All the missing proofs of results quoted here can be found in these works as well as the books referred to before. The reader is assumed to be familiar with calculus on manifolds and the ele- mentary theory of Lie groups and Lie algebras, as found in e.g. [AbMa78], [AMR88], [DFN95], [Jost], [Lang], [MaRa94], [Sp79], [Serre], or [W83]. [Bou71, Bou89] is always helpful when a quick recall of the statement of a theorem is needed. 1.1 Lagrangian and Hamiltonian Formalism Let us start with Newton’s equations for N particles q := (q1,...,qN ) ∈ 3N R with masses m1,...,mN ∈ R.IfF =(F1, ..., FN ) are the forces acting on these particles then Newton’s equations are ma = F, (1.1) where a = q¨ is the acceleration of the system. Assuming that the forces are induced by a potential V : R3N → R,thatis, F(q)=−∇V (q), (1.2) equations (1.1) become ∂V miq¨i = − ,i=1,...,N, (1.3) ∂qi where ∂V/∂qi denotes the gradient relative to the variable qi. Astraightforward verification shows that if one defines the Lagrangian function L : R6N = {(q, q˙ ) | q, q˙ ∈ R3N }→R by 1 N L(q, q˙ ):= m q˙ 2 − V (q). (1.4) 2 i i i=1 and assumes that q˙ = dq/dt,then Newton’s equations (1.2) are equivalent to Lagrange’s equations d ∂L ∂L − =0,i=1,...,N, (1.5) dt ∂q˙ i ∂qi V. 1 Introduction 3 3 3 where ∂L/∂q˙ i,∂L/∂qi ∈ R denote the gradients in R of L relative to 3 q˙ i, qi ∈ R . On other hand, the Lagrange equations (1.5) are equivalent with the varia- tional principle of Hamilton: the solutions of (1.5) are critical points of the action functional defined on the space of smooth paths with fixed endpoints. More precisely, let Λ([a, b], R3N ) be the space of all possible smooth trajecto- 3N ries q :[a, b] → R with fixedendpoints qa = γ(a), qb = γ(b).Theaction functional is defined by: b A[q(·)] := L(q(t), q˙ (t)) dt, (1.6) a where q˙ = dq(t)/dt.InΛ([a, b], R3N ) consider a deformation q(t, s), s ∈ (−, ), >0,with fixedendpoints qa, qb,ofacurve q0(t),thatis,q(t, 0) = q0(t) for all t ∈ [a, b] and q(a, s)=q0(a)=qa, q(b, s)=q0(b)=qb for 3N all s ∈ (−, ).Define a variation of the curve q0(·) in Λ([a, b], R ) by d · · ∈ R3N δq( ):= q( ,s) Tq0(·)Λ([a, b], ), ds s=0 and the first variation of A at q0(t) to be the following derivative: d DA[q0(·)](δq(·)) := A[q(·,s)]. (1.7) ds s=0 Note that δq(a)=δq(b)=0.With these notations, the variational principle of Hamilton states that the curve q0(t) satisfies the Lagrange equations (1.5) if and only if q0(·) is a critical point of the action functional, that is, DA[q0(·)] = 0. Indeed, using the equality of mixed partials, integrating by parts, and taking into account that δq(a)=δq(b)=0,weget b d d DA[q0(·)](δq(·)) = A[q(·,s)] = L(q(t, s), q˙ (t, s)) dt ds ds s=0 s=0 a b ∂L ∂L = δq(t, s)+ δq˙ dt ∂q ∂q˙ i a i i b d ∂L ∂L = − − δqidt =0 a dt ∂q˙ i ∂qi for all smooth δqi(t) satisfying δqi(a)=δqi(b)=0,which proves the claim. Next, introduce the conjugate momenta i ∂L 3 p := = miq˙ i ∈ R ,i=1,...,N. (1.8) ∂q˙ i 4 VACrash Course in Geometric Mechanics Define the change of variables (q, q˙ ) → (q, p), called the Legendre trans- form,andthe Hamiltonian H(q, p):=p · q˙ (q, p) − L(q, q˙ (q, p)) 1 N = m q˙ 2 + V (q) 2 i i i=1 1 N 1 = pi2 + V (q) (1.9) 2 m i=1 i which is the total energy of the system, expressed in the variables (q, p).Then one has ∂H 1 dq i ˙ i i = p = qi = ∂p mi dt and ∂H ∂V ∂L = = − . ∂qi ∂qi ∂qi Therefore, by the Lagrange equations (1.5) we have dpi d ∂L ∂L ∂H p˙ i = = = = − . dt dt ∂q˙ i ∂qi ∂qi This shows that Lagrange’s equations (1.5) are equivalent to Hamilton’s equa- tions ∂H ∂H ˙ ˙ i − qi = i p = , (1.10) ∂p ∂qi i 3 where, as before, ∂H/∂qi,∂H/∂p ∈ R are the gradients of H relative to i 3 qi, p ∈ R . Note that, whereas Lagrange’s equations are of second order and concern curves in the configuration space (the space of q’s), Hamilton’s equations are of first order and describe the dynamics of curves belonging to phase space, aspace of dimension twice the dimension of the configuration space whose points are pairs formed by configurations q and conjugate momenta p. An easy verification shows that Hamilton’s equations (1.10) can be equiva- lently written as F˙ = {F, H} for all F ∈F(P ), (1.11) where F(P ) denotes the smooth functions on the phase space P := R6N ,and V. 1 Introduction 5 the Poisson bracket is defined by N ∂G ∂K ∂G ∂K {G, K} := · − · for all G, K ∈F(P ). ∂q ∂pi ∂pi ∂q i=1 i i (1.12) Indeed, from (1.12) and (1.10) we have, for any F ∈F(P ), N ∂F ∂F dF · q˙ + · p˙ i = ∂q i ∂pi dt i=1 i N ∂F ∂H ∂F ∂H = {F, H} = · − · ∂q ∂pi ∂pi ∂q i=1 i i which is equivalent to (1.10) since F ∈F(P ) is arbitrary. Summarizing, for classical mechanical systems in Euclidean space describ- ing particle motion, whose total energy is given by kinetic plus potential en- ergy, we have shown that Newton’s equations are equivalent to: • Lagrange’s equations • Hamilton’s variational principle • Hamilton’s equations of motion • Hamilton’s equations in Poisson bracket formulation. In the course of these lectures we shall focus on each one of these four pictures and shall explain the geometric structure underlying them when the configuration space is a general manifold. It turns out that, in general, they are not equivalent and, moreover, some of these formulations have very useful generalizations, particularly appropriate for systems with symmetry, the case we shall consider next by means of an example. 1.2 The Heavy Top In these lectures we shall discuss the equivalences just described in the context of systems with symmetry when one can eliminate variables.
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