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 $H$ iscalled 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 Hamiltonianvector 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 canonicallocal 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.