Symplectic Integrators: Störmer-Verlet vs Euler

Symplectic Integrators: A numerical method is symplectic if it preserves the symplectic 2-form ω = dq∧dp — the geometric structure of Hamiltonian mechanics. Liouville's theorem: phase space volume is conserved (det(J)=1 for the flow map).

Störmer-Verlet (leapfrog): p_{n+1/2} = p_n − (h/2)∇V(q_n); q_{n+1} = q_n + h·p_{n+1/2}/m; p_{n+1} = p_{n+1/2} − (h/2)∇V(q_{n+1}). This is second-order and exactly symplectic. It conserves a shadow Hamiltonian H̃ = H + O(h²) — energy oscillates but does NOT drift.

Euler methods: Forward Euler is NOT symplectic — its Jacobian has det > 1, causing volume expansion → energy grows without bound. Backward Euler contracts phase space → energy decays. Only symplectic-Euler (one position, one momentum update) is symplectic.

Left: Phase space portrait (closed orbits = energy conservation). Right: Energy error ΔE/E₀ vs time — Verlet oscillates O(h²), Euler drifts linearly.