Symplectic Integrator — Kepler Orbit

Eccentricity e 0.60 Time step Δt 0.03 Trail length 300

● Cyan: Störmer-Verlet (symplectic)

● Orange: Forward Euler

Störmer-Verlet / leapfrog is a symplectic integrator — it preserves the symplectic 2-form dq∧dp exactly, keeping the orbit on an invariant torus. Energy oscillates but never drifts. The update is:
p_n+1/2 = p_n + (Δt/2)F(q_n), q_n+1 = q_n + Δt·p_n+1/2, p_n+1 = p_n+1/2 + (Δt/2)F(q_n+1)
Forward Euler (non-symplectic) spirals outward — area in phase space is not preserved and energy grows secularly. The energy error plot shows oscillating (symplectic) vs. drifting (Euler) behavior clearly.

Symplectic geometryStörmer-VerletKepler problemEnergy conservationHamiltonian mechanics