Hamiltonian vector field

Let (M,ω) be a symplectic manifoldMathworldPlanetmath, and ω~:TMT*M be the isomorphism from the tangent bundleMathworldPlanetmath to the cotangent bundleMathworldPlanetmath


and let f:M is a smooth function. Then Hf=ω~-1(df) is the Hamiltonian vector field of f. The vector field Hf 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 ω~(X)=ω(,X) is exact.

If T*Q is the cotangent bundle of a manifoldMathworldPlanetmath 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 Hf 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