Hamilton equations

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

Local formulation

Suppose $U\subseteq\mathbbm{R}^{n}$ is an open set, suppose $I$ is an interval (representing time), and $H\colon U\times\mathbbm{R}^{n}\times I\to\mathbbm{R}$ is a smooth function. Then the equations

	$\displaystyle\dot{q}_{j}$	$\displaystyle=\frac{\partial H}{\partial p_{j}}(q(t),p(t),t),$		(1)
	$\displaystyle\dot{p}_{j}$	$\displaystyle=-\frac{\partial H}{\partial q_{j}}(q(t),p(t),t),$		(2)

are the Hamilton equations for the curve

(q,p)=(q_{1},\ldots,q_{n},p_{1},\ldots,p_{n})\colon I\to U\times\mathbbm{R}^{n}.

Such a solution is called a bicharacteristic, and $H$ is called a Hamiltonian function. Here we use classical notation; the $q_{i}$ ’s represent the location of the particles, the $p_{i}$ ’s represent the momenta of the particles.

Global formulation

Suppose $P$ is a symplectic manifold with symplectic form $\omega$ and that $H\colon P\to\mathbbm{R}$ is a smooth function. Then $X_{H}$ , the Hamiltonian vector field corresponding to $H$ is determined by

dH=\omega(X_{H},\cdot).

The most common case is when $P$ is the cotangent bundle of a manifold $Q$ equipped with the canonical symplectic form $\omega=-d\alpha$ , where $\alpha$ 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 $X_{H}$ . Given a system of coordinates $x^{1},\ldots x^{2n}$ on the manifold $P$ , they can be written as follows:

\dot{x}^{i}=(X_{H})^{i}(x_{1},\ldots x_{2n},t)

The relation with the former definition is that in canonical local coordinates $(q_{i},p_{j})$ for $T^{\ast}Q$ , the flow of $X_{H}$ 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 product $P\times\mathbbm{R}$ 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