= Hamiltonian isotopy
{c}
{title2=$\phi_t$}
A Hamiltonian isotopy is a smooth path of <symplectomorphisms> $\phi_t$, with $\phi_0=\mathrm{id}$, generated by a smooth time-dependent <Hamiltonian function>. In the convention $\iota_{X_{H_t}}\omega=-dH_t$, it satisfies
$$
\frac{d}{dt}\phi_t=X_{H_t}\circ\phi_t.
$$
The flow must exist throughout the chosen time interval. This is automatic on a closed <symplectic manifold>; on an open one, uniformly compactly supported Hamiltonians also suffice. The <Hamiltonian flow preserves the symplectic form> by <Cartan's magic formula>.
Its time-one map is a <Hamiltonian diffeomorphism>, an element of the <group of Hamiltonian diffeomorphisms>. A path and its endpoint are different data: a full rotation of the <symplectic plane> gives a nonconstant Hamiltonian path whose endpoint is the identity. Distinct paths can therefore represent the same <Hamiltonian diffeomorphism>.
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