Let $(M,\omega)$ be a symplectic manifold, and $\tom:TM\to T^*M$ be the isomorphism from the tangent bundle to the cotangent bundle $$X\mapsto\om(\cdot,X)$$ and let $f:M\to\R$ is a smooth function. Then $H_f=\tom^{-1}(df)$ is the Hamiltonian vector field of $f$ . The vector field $H_f$ is symplectic, and a symplectic vector field $X$ is Hamiltonian if and only if the 1-form $\tom(X)= \om(\cdot,X)$ is exact.

If $T^*Q$ is the cotangent bundle of a manifold $Q$ , which is naturally identified with the phase space of one particle on $Q$ , and $f$ is the Hamiltonian, then the flow of the Hamiltonian vector field $H_f$ is the time flow of the physical system.