What is Hamiltonian and canonical equations? — Physics

Definition 2.1 University Physics — Year 3 · Chapter 2 — Hamiltonian Mechanics

For a system with Lagrangian $L(q, \dot q, t)$ , express the velocities in terms of the momenta $p_i = \partial L/\partial\dot q_i$ and define the Hamiltonian as the Legendre transform

H(q, p, t) = \sum_i p_i\dot q_i - L ,

a function of the coordinates and the momenta. The $2n$ -dimensional space of the $(q_1, \dots, q_n, p_1, \dots, p_n)$ is the phase space; one point of it — one state — determines the entire future and past of the system through the canonical equations of Theorem 2.2.

Examples

Example 2.4 (Two Hamiltonians)

Mass on a spring: $p = m\dot x$ , so

H = \frac{p^2}{2m} + \frac12 kx^2 ;

the canonical equations $\dot x = p/m$ , $\dot p = -kx$ are the familiar pair. Pendulum: $p = m\ell^2\dot\theta$ and

H = \frac{p^2}{2m\ell^2} - mg\ell\cos\theta .

In both cases $H$ is the energy, constant on each motion: the motions are the level curves of $H$ in the $(q,p)$ plane.

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Examples

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