Phase space volume is conserved under Hamiltonian flow — the ensemble of trajectories moves like an incompressible fluid. Works for harmonic oscillator, anharmonic oscillator, and pendulum.
Oscillator
Area: —
Liouville's Theorem
∂ρ/∂t + {ρ,H} = 0
{·,·} = Poisson bracket. Phase space density ρ is conserved along trajectories.
Consequence: cannot focus a beam in phase space beyond its initial density (basis of Liouville's theorem in optics and beam physics).