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.
