Conservation of Geometry in a Two-Body Newtonian System Using Different Integrators

The two-body problem in Newtonian gravity is exactly solvable and conserves both mechanical energy and angular momentum, which fix the shape and orientation of the orbit. When solving differential equations like the Kepler problem, there is often a tradeoff between conservation of the geometry of the system and its accuracy, as different integrators conserve quantities to different fidelity. We simulate the Earth-Sun system for 100 orbital periods using four different integrators: Explicit Euler, Semi-implicit Euler, Leapfrog, and Runge-Kutta 4. We track conserved quantities such as relative angular momentum error and total mechanical energy error alongside phase-space portraits and orbital trajectories. Explicit Euler shows a monotonic energy increase and spirals outward. RK4, despite being fourth-order accurate, exhibits secular energy drift and angular momentum growth. The symplectic integrators bound energy and conserve angular momentum to near machine precision. Trajectory plots and phase-space portraits corroborate these results. We conclude that symplecticity, not formal order of accuracy, is the critical property for long-term geometric fidelity in orbital simulations.