Example: 1‑D Harmonic Oscillator (mass on a spring) 1. Write the Hamiltonian: H = p²/(2m) + ½kq². 2. Hamilton’s equations: dq/dt = ∂H/∂p = p/m dp/dt = -∂H/∂q = -kq 3. Solve for the trajectory in phase space: differentiate dq/dt to get d²q/dt² = - (k/m) q, which is the simple harmonic equation. 4. The solution is q(t) = A cos(ωt) + B sin(ωt) with ω = √(k/m). The corresponding momentum is p(t) = m dq/dt. 5. Eliminating time gives p²/(2m) + ½kq² = E (constant energy). This is an ellipse in (q, p) space. 6. Liouville’s theorem: The area inside this ellipse, A = π√(2mE/k), stays constant as the system evolves. Even if the ellipse stretches along one axis, it compresses along the other so that the total area is unchanged.