Defining canonical momenta $p_i=\partial L/\partial \dot{q}_i$ allows one to construct the Hamiltonian $H(q,p,t)=\sum_i p_i\dot{q}_i-L$. When the Legendre transform is regular, Hamilton’s equations $\dot{q}_i=\partial H/\partial p_i$ and $\dot{p}_i=-\partial H/\partial q_i$ replace second-order equations with a first-order flow on phase space.

The canonical phase space carries a symplectic form $\omega=\sum_i dq_i\wedge dp_i$. Hamiltonian flow preserves this structure and the phase-space volume element, $d\Gamma=\prod_i dq_i\,dp_i$, expressing Liouville’s theorem. This has profound consequences for statistical mechanics, chaos, and numerical methods.

Canonical transformations preserve the symplectic structure and can simplify the Hamiltonian, expose conserved quantities, or enable perturbation theory. Generating functions provide a systematic way to construct such transformations, and action-angle variables are especially powerful for integrable systems.