Hamiltonian diffeomorphism

Let $(M,\omega)$ be a symplectic manifold. A Hamiltonian diffeomorphism is a diffeomorphism $\phi:M\to M$ for which there is a smooth Hamiltonian $H:[0,1]\times M\to\mathbb R$ whose time-dependent Hamiltonian flow $\phi_H^t$ exists for every $t\in[0,1]$ , begins at $\phi_H^0=\operatorname{id}_M$ , and satisfies $\phi=\phi_H^1$ . Equivalently, $\phi$ is the endpoint of a Hamiltonian isotopy.

On a noncompact $M$ , the compactly supported group convention used here requires the family $H_t$ to have support in one fixed compact subset of $M$ . This is stronger and less ambiguous than requiring only its vector field or time-one map to be compactly supported. When needed, the notation $\operatorname{Ham}_c(M,\omega)$ makes this convention explicit.

Group structure

Hamiltonian diffeomorphisms form a normal subgroup

\operatorname{Ham}(M,\omega)\trianglelefteq\operatorname{Symp}_0(M,\omega)

of the identity component of the symplectomorphism group. Reversing a Hamiltonian isotopy gives the inverse, and concatenating suitably reparametrized isotopies gives the product. Conjugating by a symplectomorphism transports the generating Hamiltonian and keeps the resulting isotopy Hamiltonian.

Relation to symplectic isotopies

Every Hamiltonian isotopy is symplectic, but a symplectic isotopy need not be Hamiltonian. For a symplectic isotopy generated by $X_t$ , the one-forms $\iota_{X_t}\omega$ are closed; Hamiltonianity asks that they be exact, coherently generated by functions $H_t$ . The flux homomorphism measures the obstruction at the level of path classes. On a closed connected manifold it descends modulo the flux group to

\overline{\operatorname{Flux}}: \operatorname{Symp}_0(M,\omega)\to H^1_{\mathrm{dR}}(M;\mathbb R)/\Gamma_\omega

with kernel $\operatorname{Ham}(M,\omega)$ . In particular, the inclusion $\operatorname{Ham}(M,\omega)\subseteq\operatorname{Symp}_0(M,\omega)$ can be strict.

Examples and non-examples

The time-one map of any complete autonomous Hamiltonian flow is Hamiltonian. On the standard symplectic torus, translation by a nonzero vector is symplectically isotopic to the identity, but the translation isotopy generally has nonzero flux and is not Hamiltonian. On a closed connected manifold, adding an arbitrary function of time to $H_t$ changes neither its vector field nor its endpoint.

Conventions and scope

Some authors define Hamiltonian diffeomorphisms only on closed manifolds and introduce $\operatorname{Ham}_c(M,\omega)$ for compactly supported maps on an open manifold. Others build compact support into the notation $\operatorname{Ham}(M,\omega)$ . The support condition should therefore be checked whenever $M$ is noncompact.

References

Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: Chapter 10, Hamiltonian diffeomorphisms and flux.
Augustin Banyaga, The Structure of Classical Diffeomorphism Groups, Kluwer Academic Publishers, 1997. DOI record. Relevant: Chapter 7, symplectic and Hamiltonian diffeomorphism groups.