Hamilton's equations

Remarks

The proof is a computation of partial derivatives of $H = p \cdot v - L(q, v, t)$, where $v(q, p, t)$ is the inverse of $\dot q \mapsto \partial L / \partial \dot q$. Since $\partial L / \partial \dot q (q, v, t) = p$, the terms with derivatives of $v$ cancel, which gives $\partial H / \partial p = v$, $\partial H / \partial q = -\partial L / \partial q$ and $\partial H / \partial t = -\partial L / \partial t$. The first equation $\dot q = \partial H / \partial p$ is then the definition of $p$, and the second is the Euler–Lagrange equation $\dot p = \partial L / \partial q$.

Example: for $L = \frac{1}{2} m \dot q^2 - \frac{1}{2} k q^2$, $H = p^2/(2m) + k q^2/2$, and Hamilton's equations $\dot q = p/m$, $\dot p = -k q$ combine to $m \ddot q = -k q$.

Without hyperregularity, $H$ need not be defined: for $L = \frac{1}{2} (\dot q_1 - q_2)^2$ the momentum $p_2$ is identically $0$, and the Legendre transform is not invertible; such systems need the theory of constraints.

Going deeper

For a student

Hamilton's equations say that the velocity in phase space is the Hamiltonian vector field of $H$. This form is the starting point for Liouville's theorem on the conservation of phase-space volume, for canonical transformations, and for the Hamilton–Jacobi equation. A quantity $f(q, p)$ evolves by $\dot f = \{f, H\} + \partial f / \partial t$, with the Poisson bracket. In canonical coordinates the equations keep the same form, which is why changes of variables that preserve this form are so useful.