# Hamiltonian vector field

Let $(M,\omega)$ be a symplectic manifold, and $\tilde{\omega}:TM\to T^{*}M$ be the isomorphism from the tangent bundle to the cotangent bundle

 $X\mapsto\omega(\cdot,X)$

and let $f:M\to\mathbb{R}$ is a smooth function. Then $H_{f}=\tilde{\omega}^{-1}(df)$ is the Hamiltonian vector field of $f$. The vector field $H_{f}$ is symplectic (http://planetmath.org/SymplecticVectorField), and a symplectic vector field $X$ is http://planetmath.org/node/6410Hamiltonian if and only if the 1-form $\tilde{\omega}(X)=\omega(\cdot,X)$ 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.

Title Hamiltonian vector field HamiltonianVectorField 2013-03-22 13:14:07 2013-03-22 13:14:07 rspuzio (6075) rspuzio (6075) 7 rspuzio (6075) Definition msc 53D05