Hamilton equations

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

Local formulation

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

${\dot{q}}_j$ $={\displaystyle \frac{\partial H}{\partial p_j}}(q(t),p(t),t),$ (1)

${\dot{p}}_j$ $=-{\displaystyle \frac{\partial H}{\partial q_j}}(q(t),p(t),t),$ (2)

are the Hamilton equations for the curve

$$(q,p)=(q_1,\mathrm{\dots },q_n,p_1,\mathrm{\dots },p_n):I\to U\times {ℝ}^n.$$

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 $H:P\to ℝ$ 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 $x^1,\mathrm{\dots }{x}^{2n}$ on the manifold $P$, they can be written as follows:

$${\dot{x}}^i={(X_H)}^i(x_1,\mathrm{\dots }{x}_{2n},t)$$

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 $P\times ℝ$ 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