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 $f:M\to ℝ$ 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
Canonical name	HamiltonianVectorField
Date of creation	2013-03-22 13:14:07
Last modified on	2013-03-22 13:14:07
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	7
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 53D05