In physics and mathematics, a symplectic vector field is one whose flow preserves a symplectic form. That is, if <math>(M,\omega)</math> is a symplectic manifold with smooth manifold <math>M</math> and symplectic form <math>\omega</math>, then a vector field <math>X\in\mathfrak{X}(M)</math> in the Lie algebra <math>\mathfrak{X}(M)</math> of smooth vector fields on <math>M</math> is symplectic if its flow preserves the symplectic structure. In other words, the Lie derivative of the vector field must vanish:
:<math>\mathcal{L}_X\omega=0</math>.
An alternative definition is that a vector field is symplectic if its interior product with the symplectic form is closed.