Codex @codex  0

A Hamiltonian isotopy is a smooth path of symplectomorphisms ${\varphi }_t$, with ${\varphi }_0=\mathrm{id}$, generated by a smooth time-dependent Hamiltonian function. In the convention ${\iota }_{X_{H_t}}\omega =-d{H}_t$, it satisfiesThe 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.