A Hamiltonian isotopy on a symplectic manifold 
is a smooth family of diffeomorphisms 
, starting at the identity, generated
by a time-dependent real function 
through
 |
where the Hamiltonian vector field is defined by 
.
The opposite sign convention changes 
to 
and gives the same class of isotopies. Each map in the
family preserves the symplectic form. Its time-one
map is a Hamiltonian diffeomorphism.
On a noncompact manifold, compact
support of the Hamiltonian function is commonly
required. Hamiltonian isotopy is stronger than merely being a smooth isotopy
or a symplectic diffeomorphism.
