= Hamiltonian diffeomorphism
{c}
A Hamiltonian diffeomorphism is the time-one map of a <Hamiltonian isotopy> beginning at the identity on a <symplectic manifold>. The isotopy is generated by a smooth time-dependent <Hamiltonian function> $H_t$. Its flow must exist for the whole interval; on an open manifold a common group convention requires compact support throughout that interval.
In the convention $\iota_{X_{H_t}}\omega=-dH_t$, the infinitesimal preservation law is
$$
\mathcal L_{X_{H_t}}\omega=d(\iota_{X_{H_t}}\omega)+\iota_{X_{H_t}}d\omega=-d^2H_t=0.
$$
Thus every Hamiltonian diffeomorphism is a <symplectomorphism>. The conclusion follows from closedness of the symplectic form, not from a special choice of coordinates.
The identity is generated by the zero Hamiltonian. Inverses and compositions are again Hamiltonian endpoints, forming the <group of Hamiltonian diffeomorphisms>. A generating function path and its endpoint are different data: different isotopies can have the same time-one map. The Hamiltonian condition imposes more structure than only preserving the symplectic form.
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