Integrator Race: Euler vs Verlet vs RK4

Four identical bodies are launched into the same elliptical orbit around the same central mass, each advanced by a different numerical integrator — Explicit Euler, Symplectic (semi-implicit) Euler, Velocity Verlet and RK4 — using the same shared step size. Because Newton's laws are differential equations with no closed-form solution for most real scenarios, every physics or orbital-mechanics engine has to pick one of these stepping schemes, and the choice matters: explicit Euler visibly spirals outward as it leaks energy every step, the two symplectic methods keep a stable closed ellipse even at a coarse step size, and RK4 trades symplecticity for high per-step accuracy. Live energy-drift readouts per integrator, a shared Δt/gravity/eccentricity control set, and per-body trails make the divergence easy to see and reason about.