Compare Euler, RK4, and Leapfrog (Störmer-Verlet) for a harmonic oscillator H = p²/2 + q²/2. Leapfrog exactly conserves a modified Hamiltonian — its phase portrait stays on a closed orbit forever.
Phase portrait (left) — Energy H(t) over time (right). Leapfrog: H stays bounded. Euler: H grows (spiral out). RK4: tiny drift.