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1 Symplectic Geometry in Classical Mechanics Aspects of Symplectic Geometry in Physics Josh Powell 1 Symplectic Geometry In Classical Mechanics We seek here to use geometry to gain a more solid understanding of physics. This process may seem frighteningly mathematical, but we shouldn’t expect much gain in insight with only a modicum of effort. Following Alexander Pope: “A little learning is a dangerous thing; Drink deep, or taste not the Pierian spring: There shallow draughts intoxicate the brain, And drinking largely sobers us again.” 1.1 Hasty Review of Classical Mechanics Consider a mechanical system with both position degrees of freedom, qi, and momentum degrees of freedom, i pi. The Hamiltonian H(pi, q ) is a function on phase space that governs the dynamics of the system, and in particular of these degrees of freedom. That is, it describes time evolution by specifying their equations of motion. The equations of motion are i ∂H ∂H q˙ = andp ˙i = − i ∂pi ∂q We have a system of differential equations which are all first order in time derivatives. Since we know the i expression for the phase space variables’ time derivatives, any function f(pi, q ) of these variables will have a time derivative given by df X ∂f ∂f = q˙i + p˙ dt ∂qi ∂p i i i X ∂f ∂H ∂f ∂H = − =: {f, H} ∂qi ∂p ∂p ∂qi i i i where in the last line we have defined the Poisson bracket of two functions on phase space. The function i also been explicitly dependent on time, f = f(pi, q , t), then we would have written df ∂f = {f, H} + dt ∂t in order to capture the total time derivative. We have thus seen how the Hamiltonian gives the time evolution of all dynamical quantities in our system. What if we had chosen another coordinate system on our phase space? Let’s call this new coordinate i i system (Pi,Q ) to distinguish from the previous (pi, q ). The dynamics in the old system are described by varying the action Z X i δ pidq − Hdt = 0 and those in the new come from varying Z X i 0 δ PidQ − H dt = 0 The same dynamics are described in terms of these two coordinate systems if these two integrands differ by i a total differential dF of some function F = F (pi, q , t). We write this as X i X i 0 dF = pidq − PidQ + (H − H)dt i i 0 whence we get ∂F/∂q = pi, ∂F/∂Q = −Pi and ∂F/∂t = H − H. We call the function F the generating function of this canonical transformation. If F has no explicity time dependence, it’s an element of F(Q×Q). If it is time dependent, then F ∈ F(Q × Q × R), and the two Hamiltonians differ in a way more than just 1 rewriting the old Hamiltonian in terms of the new coordinates. There is another common formalism in which the dynamics of a system can be described. Imagine we had i i i i wanted to work not in terms of the variables (pi, q ) but instead (q , q˙ )? Since ∂H/∂pi =q ˙ , we recognize i that what this amounts to is a Legendre transform from pi toq ˙ . Let’s say that the Legendre transform of i i i the Hamiltonian H(pi, q ) is some function L(q , q˙ ). This transformation and its inverse are given by i ˙i X i i i L(q , q ) = piq˙ − H(pi, q ) with pi = ∂L/∂q˙ i X i i i i H(pi, q ) = piq˙ − L(q , q˙ ) withq ˙ = ∂H/∂pi i i i The first canonical equation of motion, ∂H/∂pi =q ˙ , is transformed into the tautological statementq ˙ =q ˙ . This comes as no surprise, since it was the equation which specified in the first place the transformation we i are making. The second canonical equation of motion, ∂H/∂q = −q˙i, is transformed into ∂L d ∂L − = 0 ∂qi dt ∂q˙i which is known as the Lagrange equation of motion. Note that it is second order in time derivatives, unlike the dynamical equations in the Hamiltonian formalism which were both first order. We’ve obtained the Lagrangian formulation of classical mechanics. The two are equivalent descriptions, up to certain technical issues related to when the Legendre transformation is not defined. 1.2 Symplectic Geometry Let M be a smooth manifold of even dimensionality and let Ω be a closed, non-degenerate 2-form on M. We call the pair (M, Ω) a symplectic manifold, and Ω its symplectic structure. The condition that Ω be non-degenerate means that the only vector field X on M satisfying Xy Ω = 0 is the one vanishing uniformly. A non-zero X satisfying this condition would correspond to a zero-mode of the symplectic structure, and would prevent Ω from being invertible. If such a zero-mode exists, it’s possible that it can be projected out and a lower dimensional system thus obtained is a symplectic manifold with non-degenerate symplectic structure. This procedure is known as reduction. One familiar example of this process comes from gauge theories, in which it is sometimes necessary to project out non-physical degrees of freedom. In this case, it’s known as gauge fixing. In general, any diffeomorphism of M which preserves the symplectic structure is known as a sym- plectomorphism. That is, a symplectomorphism is a diffeomorphism between two symplectic manifolds ∗ f :(M, ΩM ) → (N, ΩN ) for which f ΩN = ΩM . We can generate diffeomorphisms (M, Ω) → (M, Ω) by exponentiating a vector field on M. If this vector field, call it X, is to generate a symplectomorphism, then it must satisfy LX Ω = 0. Vector fields satisfying this condition are known as symplectic vector fields. Furthermore, given a function f ∈ F(M), there is Xf ∈ X(M) satisfying Xf y Ω + df = 0 (∗) The invertibility of Ω guarantees the existence of Xf so defined. In local coordinates, this vector field has a ab the components (Xf ) = Ω ∂bf. Such a vector field is automatically symplectic: LXf Ω = Xf y dΩ + d Xf y Ω = d(−df) = 0 We call the function f the generating function and the vector field Xf a Hamiltonian vector field. Conversely, given a symplectic vector field X, we can try to find the corresponding generating function by working back- wards. We can of course do this locally, but in order to find a globally defined function that generates X, 2 we need the first de Rham group to be trivial H1(M) = 0. Therefore, obstructions which keep a symplectic vector field from being Hamiltonian arise from H1(M). Requiring that dΩ = 0 allows one to write Ω = dΘ, at least locally. Depending on the topology of M, this may hold for a globally define 1-form Θ. Θ is known as the symplectic potential. Imagine we have a Hamiltonian vector field Xf generated by f. What is the change in Θ under an infinitessimal flow along Xf ? LXf Θ = Xf y dΘ + d Xf y Θ = Xf y Ω + d Xf y Θ = d Θ(Xf ) − f =: dΛ where we used the relation (∗). Then Θ is an abelian gauge potential and Ω = dΘ is the corresponding field strength. From this point of view, the flow Ft(Xf ): M → M which preserves Ω but sends Θ → Θ + dΛ is a (symplectomorphic) gauge transformation. 1.2.1 Poisson Brackets If we have two Hamiltonian vector fields Xf and Xg, their Lie bracket is also Hamiltonian. L[Xf ,Xg ]Ω = [LXf , LXg ]Ω = 0 What is the corresponding generating function? [Xf ,Xg]y Ω = (LXf Xg)y Ω = LXf (Xgy Ω) − Xgy (LXf Ω) = Xf y d Xgy Ω + d Ω(Xg,Xf ) = −d Ω(Xf ,Xg) Therefore, we define the Poisson bracket of the functions f and g as {f, g} := Ω(Xf ,Xg) [Xf ,Xg]y Ω + d{f, g} = 0 We could have stopped midway in our derivation of the Poisson bracket and ended up with other equivalent expressions: {f, g} = df(Xg) = LXg f and also {f, g} = −dg(Xf ) = −LXf g. In local coordinates, the Pois- −1 ab a b son bracket takes the form {f, g} = (Ω ) ∂af∂bg. Note in particular that if ξ and ξ are two coordinate functions on phase space, we have {ξa, ξb} = (Ω−1)ab. The salient features of the Poisson bracket – namely, anti-symmetry and the Jacobi identity – follow readily from the definition. - Anti-symmetry: {g, f} = Ω(Xg,Xf ) = −Ω(Xf ,Xg) = −{f, g} - Jacobi Identity: From the second boxed equation above, we see that X{f,g} = −[Xf ,Xg]. From here, we find {f, {g, h}} = LX{g,h} f = −L[Xg ,Xh]f = −[LXg , LXh ]f = LXh (LXg f) − LXg (LXh f) = LXh {f, g} − LXg {f, h} = {{f, g}, h} + {g, {f, h}} 3 Hence, the Poisson bracket and its properties are entirely geometric notions defined on any symplectic man- ifold. Let Xf be a Hamiltonian vector field on M with generating function f. Then, Xf generates a one- parameter family of diffeomorphisms, its flow, through exponentiation, and is the first-order term in the expansion of an infinitessimal flow. The corresponding infinitessimal change in a function G ∈ F(M) under this infinitessimal diffeomorphism is δG = LXf G = {G, f}. 1.2.2 Theorem of Darboux Any two symplectic manifolds of the same dimension are locally symplectomorphic, so all their distinguishing features must manifest as global properties. These are the famed Gromov-Witten invariants. Compare this situation to Riemannian geometry, where symmetries of the manifold must preserve the Riemann curvature tensor.
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