Pendulum Phase Space

Physics: Hamiltonian H = ½ml²θ̇² − mgl·cosθ. Separatrix at H = mgl (energy to reach unstable equilibrium at θ = π). Below separatrix: libration (closed orbits, pendulum oscillates). Above: rotation (open orbits, pendulum spins). Homoclinic orbit connects the saddle points at (±π, 0) — infinite period. Hamiltonian systems obey Liouville's theorem: phase-space volume is preserved. With damping, orbits spiral inward to stable fixed point at (0, 0).