Hamiltonian isotopy - Knowlpedia

Definition

Let $(M,\omega)$ be a symplectic manifold. A Hamiltonian isotopy is a symplectic isotopy $\phi_t$ , beginning at $\phi_0=\operatorname{id}_M$ , whose generating vector field $X_t$ satisfies

\iota_{X_t}\omega=dH_t

for a smooth family of functions $H_t:M\to\mathbb R$ . Equivalently, $\phi_t$ is the evolution of the time-dependent Hamiltonian $H(t,x)=H_t(x)$ , using the corpus sign convention $\iota_{X_H}\omega=dH$ .

Each contraction form $\iota_{X_t}\omega$ is exact, not merely closed. The time-one map $\phi_1$ is a Hamiltonian diffeomorphism, and every Hamiltonian diffeomorphism arises this way.

Choice and normalization of the Hamiltonian

On a connected manifold, two Hamiltonians generate the same isotopy exactly when they differ by a function of time alone. If $M$ is closed, one may select a unique representative by imposing the mean-zero normalization

\int_M H_t\,\frac{\omega^n}{n!}=0

for every $t$ . Normalization chooses a generator; it is not an extra condition on the isotopy itself.

Noncompact support convention

For the compactly supported Hamiltonian group on a noncompact manifold, this corpus requires

\bigcup_{t\in[0,1]}\operatorname{supp}(H_t)

to lie in a compact subset of $M$ . This implies that $X_t$ and the isotopy are compactly supported. Requiring only $X_t$ , or only the endpoint, to have compact support can be weaker when constants on different ends matter, so those alternatives should not be silently substituted.

Flux characterization

A symplectic isotopy has zero pointwise cohomology class $[\iota_{X_t}\omega]$ for every $t$ exactly when it is Hamiltonian as parameterized. More generally, the vanishing of its integrated flux says that its path class relative to endpoints has a Hamiltonian representative. The distinction between a chosen path and its endpoint is essential.

References

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