Hamiltonian system

Definition

Let $n \ge 1$, let $U$ be an open subset of $\mathbb{R}^n \times \mathbb{R}^n$ with coordinates $(q, p) = (q_1, \ldots, q_n, p_1, \ldots, p_n)$, and let $H : U \to \mathbb{R}$ be continuously differentiable. The Hamiltonian system with Hamiltonian $H$ is the autonomous system $$q_i' = \frac{\partial H}{\partial p_i}(q, p), \qquad p_i' = -\frac{\partial H}{\partial q_i}(q, p), \qquad i = 1, \ldots, n.$$

Remarks

The function $H$ is a first integral when it is not constant on an open set: along a solution, $\frac{d}{dt} H = \sum_i \left( \frac{\partial H}{\partial q_i} q_i' + \frac{\partial H}{\partial p_i} p_i' \right) = 0$. For $H(q, p) = \frac{1}{2} p^2 + V(q)$ with $n = 1$, the system is $q' = p$, $p' = -V'(q)$. This is Newton's law $q'' = -V'(q)$ for a particle with potential energy $V$. The harmonic oscillator has $V(q) = \frac{1}{2} q^2$, and the pendulum has $V(q) = -\cos q$ in suitable units. A time-dependent Hamiltonian $H(t, q, p)$ gives a non-autonomous Hamiltonian system, and then $H$ need not be conserved.

Explanations

L1 human

A Hamiltonian system describes a mechanical system without friction by one function, the total energy $H$. The positions $q$ and the momenta $p$ change according to the partial derivatives of $H$, and the energy stays constant.