Hamilton equations

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

Local formulation

Suppose is an open set, suppose $I$ is an interval (representing time), and is a smooth function. Then the equations
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 $q_i$ '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 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. (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 is known as state space.