Symplectic Integrator — Störmer-Verlet

Symplectic integrators preserve the symplectic 2-form dq∧dp, ensuring exact conservation of phase-space volume (Liouville). The Störmer-Verlet (leapfrog) method: q_{n+1/2} = q_n + (h/2)p_n/m; p_{n+1} = p_n − h∇V(q_{n+1/2}); q_{n+1} = q_{n+1/2} + (h/2)p_{n+1}/m. Unlike Euler (nonsymplectic), Verlet's energy error is bounded and oscillates rather than drifting — critical for N-body simulations, molecular dynamics, celestial mechanics.