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1 Preface On August 15, 2012, I received the following message from Ryan Budney (ryan.budney@gmail.com). Hi Dick, Here s an MO post that s right up your alley. :) When I went to the MathOverflow website and looked at the relevant question, here is what I saw: In Cushman and Bates, Global Aspects of Classical Integrable Systems, 1997, I have read: In a widely circulated but unpublished letter in 1965, Palais explained the symplectic formulation of Hamiltonian mechanics. I would like to know if, in the meanwhile, this letter was made available. Giuseppe Tortorella That sixty-year old letter had in fact not been made publicly available, but I was able to locate the pages that made it up and I scanned them and made the scan available online (with an added first-page from Abraham and Marsden s Foundations of Mechanics, giving the first reference in print to that letter that I am aware of). I was very surprised at the reception this response to Giuseppe s question received within two days it got seventy-five up-votes far more than for any of my previous replies to MO questions. A little later, Giuseppe offered to make a L A TEX version of those scanned pages, and I was very happy to agree. He did an excellent job, and what follows is in order: (i) the MO page with Giuseppe s question, (ii) the original scan that I made, and (iii) Giuseppe s L A TEX version. Dick Palais 1

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8 On the Symplectic Formulation of Hamiltonian Mechanics Richard S. Palais Definition. A simplectic manifold is a C manifold M 2n together with a closed 2-form Ω of rank 2n (i.e., Ω(X, Y ) is a non-singular (alternating) bilinear form on each M p.) Lemma. Given p M, there exists a coordinate system x 1,..., x 2n near p such that Ω = n i=1 dx i dx n+i. Proof. Ω = dω locally and by classical result on one-forms ω = n i=1 x idx n+i locally. Such coordinates are called canonical this shows that all symplectic manifolds are locally equivalent. If L is a vector field on M, θ a p-form on M, we write ι L θ for contraction of θ with L, i.e., ι L θ is the (p 1)-form given by ι L θ(x 1,..., X p 1 ) = θ(l, X 1,..., X p 1 ). We write L θ for the Lie derivative of θ w.r.t. L. We recall L θ = ι L dθ + dι L θ, i.e., L = ι L d + dι L. If L is a vector field we write L = ι L Ω, a one-form on M. The fact that Ω is non-singular implies that L L is a vector space isomorphism between vector fields and one-forms. We write θ θ for the inverse isomorphism. Remark. If x 1,..., x 2n are canonical coordinates and L = X i x i then n L = X i+n dx i + i=1 n X i dx i+n. i=1 Example. Let M be a Riemannian manifold, P the bundle of covariant vectors over M, and Π : P M the usual projection. Define a one-form ω 1

9 on P by ω θ = δπ (θ), put Ω = dω. If x 1,..., x n are coordinates on O M, get coordinates x 1,..., x n, p 1,..., p n on π 1 (O) by { xi (θ) = x i (π(θ)) θ = p i (θ)dx i i.e., x i gives coordinates of base point relative to x 1,..., x n and p i gives i th component of θ relative to x 1,..., x n. It is trivial to check that ω = n i=1 p id x i, hence Ω = dω = d x i dp i, which shows that Ω has rank 2n. Therefore P is a symplectic manifold and x 1,..., x n, p 1,..., p n are canonical coordinates. This symplectic manifold is called the phase space of M. Definition. A vector field L is symplectic if L Ω = 0. (If M is compact, or more generally if L generates a global one-parameter group φ t, this is equivalent to the φ t being symplectic transformations, i.e., preserving Ω.) Lemma. L Ω = dl. Proof. Recall that L = ι L d + dι L. Since dω = 0, then L Ω = ι L dω + dι L Ω = dl. Theorem. L is symplectic if and only if L is closed, in other words the isomorphism L L between vector field and one-forms restricts to an isomorphism of symplectic vector fields and closed one-forms. Remark. Let L be a symplectic vector field, x 1,..., x n, p 1,..., p n a canonical coordinate system (with disc domain), then n ( H L = dh = dx i + H ) dp i. x i p i i=1 (H is called a local hamiltonian for L and is determined up to an additive constant.) Therefore L = n i=1 H p i x i n i=1 H, x i p i and the differential equations for integral curves of L are dx i dt = H p i (Hamiltonian system) dp i dt = H x i 2

10 Conversely of course if differential equations corresponding to L take this form then locally L = dh, so L is closed and therefore L is symplectic. i.e., Theorem. A vector field is symplectic if and only if it corresponds to a Hamiltonian system of differential equations in each canonical coordinate system. Remark. Note that a symplectic manifold has natural forms of degree 2k, (k = 1, 2,..., n), namely Ω k = Ω Ω, (k times). In particular it has a natural volume element Ω n. Note that in canonical coordinates Ω n = n!( 1) [n/2] dx 1 dx 2 dx 2n. Since L (θ ψ) = L θ ψ + θ L ψ, it follows that if L is symplectic then L Ω n = 0, i.e., a symplectic vector field generates a volume preserving one-parameter group. Then Theorem (Liouville s Theorem). If a system of differential equations in phase space P are in Hamiltonian form in canonical coordinates then they determine a volume preserving one-parameter group of transformations of P. Remark. In general if Φ is a form then the set of vector fields L such that L Φ = 0 is a Lie algebra, therefore in particular the bracket [L 1, L 2 ] of two symplectic vector fields is symplectic, and hence [L 1, L 2 ] is closed. (Note this gives a Lie algebra structure to closed forms on a symplectic manifold.) For some mysterious 1 reason [L 1, L 2 ] turns out to be exact, i.e., if S represents symplectic vector fields then the derived algebra [S, S ] gets mapped into exact forms by natural isomorphism. If it is onto then S /[S, S ] (which corresponds to abelianized group of symplectic transformations) is isomorphic to H 1 (M, R). 1 In fact [L 1, L 2 ] = d(ω(l 1, L 2 )). 3