Hamilton equations

The Hamilton equations are a formulation of the equations of motion in classical mechanics.

Local formulation

Suppose Un is an open set, suppose I is an interval (representing time), and H:U×n×I is a smooth function. Then the equations

q˙j =Hpj(q(t),p(t),t), (1)
p˙j =-Hqj(q(t),p(t),t), (2)

are the Hamilton equations for the curve


Such a solution is called a bicharacteristic, and H is called a Hamiltonian function. Here we use classical notation; the qi’s represent the location of the particles, the pi’s represent the momenta of the particles.

Global formulation

Suppose P is a symplectic manifoldMathworldPlanetmath with symplectic form ω and that H:P is a smooth function. Then XH, the Hamiltonian vector field corresponding to H is determined by


The most common case is when P is the cotangent bundle of a manifoldMathworldPlanetmath Q equipped with the canonical symplectic form ω=-dα, where α is the Poincaré 1-form (http://planetmath.org/Poincare1Form). (Note that other authors may have different sign convention.) Then Hamilton’s equations are the equations for the flow of the vector field XH. Given a system of coordinates x1,x2n on the manifold P, they can be written as follows:


The relationMathworldPlanetmathPlanetmath with the former definition is that in canonical local coordinates (qi,pj) for TQ, the flow of XH is determined by equations (1)-(2).

Also, the following terminology is frequently encountered — the manifold P is known as the phase space, the manifold Q is known as the configuration space, and the productPlanetmathPlanetmath P× is known as state space.

Title Hamilton equations
Canonical name HamiltonEquations
Date of creation 2013-03-22 14:45:58
Last modified on 2013-03-22 14:45:58
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 53D05
Classification msc 70H05
Related topic Quantization