symplectic vector field
For symplectic manifolds
For a symplectic manifold, a vector field on is called a symplectic vector field if its Lie derivative preserves the symplectic form:
For -plectic manifolds
The analogous definition applies to n-plectic geometry.
Relation to symplectomorphisms
The flow generated by a symplectic vector field is an auto-symplectomorphism.
Relation to Hamiltonian vector fields
By Cartan's magic formula and using that is by definition a closed form, the equation is equivalent to
hence equivalent to the condition that the contraction of in is a closed differential form?. If this contraction even is an exact differential form? in that there is a function such that
then the symplectic vector field is called a Hamiltonian vector field and is called its Hamiltonian function.
Created on February 26, 2012 16:29:56
by Urs Schreiber