The notion of symplcetic gradient is the analog in symplectic geometry of the gradient in Riemannian geometry.
Let be a symplectic manifold and a function.
The symplectic gradient of is the vector field
where is the de Rham differential.
This is the unique vector field such that
The function in this context is called an Hamiltonian and the vector field an Hamiltonian vector field.
Equivalently, the vector field is defined by the condition
for any , where is the Poisson bracket on .
If is endowed with the standard symplectic form , then
Revised on August 31, 2011 20:02:42
by Urs Schreiber