Symplectomorphisms and Hamiltonian Diffeomorphisms

The group of symplectomorphisms of a symplectic manifold is the natural symmetry group in symplectic geometry. The subgroup of Hamiltonian diffeomorphisms plays a special role, as these are the symmetries generated by Hamiltonian flows.

Symplectomorphism Group

Definition4.4Symplectomorphism

A symplectomorphism of $(M, \omega)$ is a diffeomorphism $\phi: M \to M$ with $\phi^*\omega = \omega$ . The group of all symplectomorphisms is denoted $\mathrm{Symp}(M, \omega)$ . Its Lie algebra consists of symplectic vector fields: $X$ with $\mathcal{L}_X \omega = 0$ , equivalently $d(\iota_X \omega) = 0$ .

Definition4.5Hamiltonian Diffeomorphism

A Hamiltonian diffeomorphism is the time-1 map $\phi_1^H$ of a (possibly time-dependent) Hamiltonian flow. The group $\mathrm{Ham}(M, \omega)$ of Hamiltonian diffeomorphisms is a normal subgroup of $\mathrm{Symp}_0(M, \omega)$ (the identity component). Its Lie algebra consists of Hamiltonian vector fields: $X_H$ where $\iota_X \omega = dH$ is exact.

ExampleFlux Homomorphism

The quotient $\mathrm{Symp}_0(M, \omega) / \mathrm{Ham}(M, \omega)$ is described by the flux homomorphism $\mathrm{Flux}: \pi_1(\mathrm{Symp}_0) \to H^1(M; \mathbb{R})$ . A symplectomorphism is Hamiltonian if and only if it can be connected to the identity by a path with zero flux. When $H^1(M) = 0$ , every symplectomorphism isotopic to the identity is Hamiltonian.

Generating Functions

Definition4.6Generating Function

A symplectomorphism $\phi$ of $T^*Q$ close to the identity can be described by a generating function $S: Q \times Q \to \mathbb{R}$ via the implicit equations $p = -\partial_1 S(q, Q)$ , $P = \partial_2 S(q, Q)$ , where $(q, p) \mapsto (Q, P) = \phi(q, p)$ .

RemarkArnold Conjecture Motivation

The fixed points of a Hamiltonian diffeomorphism $\phi$ correspond to critical points of an action functional. The Arnold conjecture predicts that $|\mathrm{Fix}(\phi)| \geq \sum_k \dim H_k(M; \mathbb{Z}_2)$ . This lower bound by topology is a deep generalization of the Poincaré-Birkhoff theorem and motivated the development of Floer homology.