Symplectic vector field
In physics and mathematics, a symplectic vector field is one whose flow preserves a symplectic form. That is, if ( M , ω ) {\displaystyle (M,\omega )} is a symplectic manifold with smooth manifold M {\displaystyle M} and symplectic form ω {\displaystyle \omega } , then a vector field X ∈ X ( M ) {\displaystyle X\in {\mathfrak {X}}(M)} in the Lie algebra X ( M ) {\displaystyle {\mathfrak {X}}(M)} of smooth vector fields on M {\displaystyle M} is symplectic if its flow preserves the symplectic structure.