Hamiltonian diffeomorphism

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 ${\iota }_{X_{H_t}}\omega =-d{H}_t$, the infinitesimal preservation law isThus 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.