Editing Symplectomorphism (section)

==Flows== <!-- [[Hamiltonian isotopy]] redirects here -->
Any smooth function on a [[symplectic manifold]] gives rise, by definition, to a [[Hamiltonian vector field]] and the set of all such vector fields form a subalgebra of the [[Lie algebra]] of [[symplectic vector field]]s. The integration of the flow of a symplectic vector field is a symplectomorphism. Since symplectomorphisms preserve the [[symplectic form|symplectic 2-form]] and hence the [[symplectic form#Volume form|symplectic volume form]], [[Liouville's theorem (Hamiltonian)|Liouville's theorem]] in [[Hamiltonian mechanics]] follows. Symplectomorphisms that arise from Hamiltonian vector fields are known as Hamiltonian symplectomorphisms.

Since {{math|1={''H'', ''H''} = ''X''<sub>''H''</sub>(''H'') = 0,}} the flow of a Hamiltonian vector field also preserves {{math|''H''}}. In physics this is interpreted as the law of conservation of [[energy]].

If the first [[Betti number]] of a connected symplectic manifold is zero, symplectic and Hamiltonian vector fields coincide, so the notions of Hamiltonian isotopy and '''symplectic isotopy''' of symplectomorphisms coincide.

It can be shown that the equations for a geodesic may be formulated as a Hamiltonian flow, see [[Geodesics as Hamiltonian flows]].