Energy conservation in numerical integration of Hamiltonian systems
Phase space (q, p)
Energy vs time
Controls
0.10
1.0
Störmer-Verlet (symplectic)
Euler (non-symplectic)
RK4 (non-symplectic)
Key Physics: Hamiltonian systems conserve energy exactly. Symplectic integrators like Störmer-Verlet preserve the symplectic structure of phase space — they conserve a modified Hamiltonian H̃ = H + O(h²), bounding energy error forever. Non-symplectic methods (Euler, RK4) exhibit secular energy drift: Euler spirals outward, RK4 spirals inward over long times. Verlet: qₙ₊₁ = 2qₙ − qₙ₋₁ + h²F(qₙ)/m; equivalent to leapfrog.