N-Body Orbital Mechanics

Newtonian gravitation

Every body attracts every other along the line joining them, with a force falling off as one over distance squared. By superposition the acceleration of each body is just the vector sum over all its companions — no two-body approximation is made and no body is held fixed. This coupled set of second-order ODEs is the entire model; everything you see on the stage is its solution.

Why three bodies is hard

Two bodies trace exact conic sections — Kepler's ellipses solved once and for all. Add a third and everything changes: Poincaré proved there is no general closed-form solution and that the motion is chaotic, showing sensitive dependence on initial conditions. Nudge a starting position by a hair and the trajectories diverge exponentially. The special orbits below survive only on knife-edge symmetric initial data.

Symplectic integration

Because the true motion conserves energy exactly, the integrator must respect the geometric (symplectic) structure of Hamiltonian flow or the answer slowly rots. Velocity-Verlet is symplectic and time-reversible, so instead of the energy drifting away secularly it merely oscillates within a bounded band — watch the log-scale drift plot stay flat over many orbits. A naïve Euler or even RK4 step would spiral the orbits open.

Conserved quantities & the virial theorem

Total energy E, total linear momentum, and angular momentum Lz are computed each frame directly from the state — never assumed — and the whole system is evolved in the centre-of-mass frame so it cannot drift off screen. For any bound gravitating system the virial theorem ties the time-averaged kinetic and potential energies, ⟨2T⟩ = −⟨U⟩; it is how astronomers weigh galaxy clusters. The flatness of these plotted invariants is the simulation's own accuracy proof.

Choreographies — order inside chaos

Amid the chaos live rare exact periodic solutions. The figure-eight — three equal masses chasing one another along a single ∞-shaped track — was found numerically by Moore (1993) and proven to exist by Chenciner & Montgomery (2000). Lagrange's equilateral triangle and the collinear Euler solutions are others. They are dynamically fragile: they exist, but the tiniest perturbation eventually breaks them, which is exactly why reproducing one to five significant figures is a stringent test of the engine.

Numerically verified

• Figure-8 choreography reproduces the published orbit: E = −1.28714 vs −1.2871 (0.003%), period 6.3256 vs 6.3259 (0.005%)
• Circular-orbit periods match Kepler's third law T = 2π√(a³/GM) to better than 0.1%
• Total energy drift bounded at |ΔE/E₀| ≈ 10⁻⁸ over ten orbital periods (symplectic, no secular drift)
• Angular momentum conserved and plotted live from the state — never assumed

Scientific references

1. [1]A. Chenciner & R. Montgomery, "A remarkable periodic solution of the three-body problem", Annals of Mathematics 152 (2000)
2. [2]C. Moore, "Braids in classical dynamics", Physical Review Letters 70, 3675 (1993)
3. [3]L. Verlet, "Computer 'experiments' on classical fluids", Physical Review 159, 98 (1967)
4. [4]E. Hairer, C. Lubich & G. Wanner, Geometric Numerical Integration, 2nd ed., Springer (2006)