Hamiltonian Mechanics
Phase space, canonical equations, Poisson brackets — the bridge to QM.
Define conjugate momenta $p_i = \partial L/\partial \dot q_i$ and the Hamiltonian $H = \sum_i p_i \dot q_i - L$. Dynamics in phase space:
$$\dot q_i = \partial H/\partial p_i, \qquad \dot p_i = -\partial H/\partial q_i.$$The Poisson bracket
$$\{f,g\} = \sum_i \left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right)$$gives $df/dt = \{f,H\} + \partial_t f$. The fundamental Poisson brackets are
$$\{q_i, p_j\} = \delta_{ij}, \qquad \{q_i, q_j\} = \{p_i, p_j\} = 0.$$Canonical (Hamiltonian) flows preserve the symplectic 2-form $\omega = dp_i \wedge dq_i$ on phase space — this is the geometric structure of classical mechanics. Dirac's quantization $\{,\} \to -i/\hbar\,[,]$ leads directly to Heisenberg-picture quantum mechanics.
Interactive: pendulum phase portrait
For $H = p^2/2 - \cos\theta$, trajectories of fixed energy are closed loops (libration) until they reach the separatrix, beyond which the pendulum rotates (circulation).