= Group of Hamiltonian diffeomorphisms
{title2=$\operatorname{Ham}(M,\omega)$}
= Hamiltonian diffeomorphism group
{c}
{synonym}
This <group> consists of time-one maps of <Hamiltonian isotopies> whose flows are defined throughout the specified time interval. With $\iota_{X_H}\omega=-dH$, composition of isotopies generated by $H_t,K_t$ has generator $H_t+K_t\circ\phi_t^{-1}$, where $\phi_t$ is the first isotopy. The inverse path has generator $-H_t\circ\phi_t$. These formulas prove closure under composition and inversion. On a closed <symplectic manifold>, every smooth time-dependent Hamiltonian has a complete flow over a bounded time interval.
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