The figure eight as a test article

§1The orbit

The general three-body problem has no closed-form solution; that is what makes it famous. But particular solutions exist, and one of the best known is the figure-eight choreography: three equal masses chasing each other around a single figure-eight path, each one third of a period apart. It was found numerically by Cristopher Moore in 1993, and proven to exist rigorously by Alain Chenciner and Richard Montgomery in 2000.

Its virtue as a bench test is exactly its rarity: it is a periodic solution. After one period, the true system returns to its starting configuration, precisely. Any disagreement between that mathematical fact and what your integrator produces is your integrator's error, made visible and measurable. Generic chaotic initial conditions cannot give you that ruler; the eight can.

§2The method

The live figure integrates Newtonian gravity with velocity Verlet at a fixed step of dt = 10⁻³, with G and all masses set to 1. Verlet is the workhorse of molecular dynamics for a reason: it is symplectic, so its energy error stays bounded and oscillatory rather than drifting steadily, which is what a naive Euler or even classic Runge-Kutta will do to you over long runs. A tiny softening term (10⁻⁹) guards the inverse-square law against floating-point trouble at close approach; at this orbit's separations it is numerically invisible.

§3What we observe

• The choreography's period is T ≈ 6.326 time units; at our step that is roughly 6,300 integration steps per lap.
• Verlet at dt 10⁻³ holds the eight visually stable for many hundreds of laps: the trace on screen re-inks the same path rather than spiralling.
• It is still not exact. Accumulated floating-point error grows slowly, and the browser figure quietly re-seeds itself every few minutes rather than pretend otherwise. That reset is an honesty policy, printed here so nobody mistakes the demo for a perpetual-motion claim.

§4What this is, and is not

This is a bench test and a beautiful object. It is not novel research; the orbit belongs to Moore, Chenciner and Montgomery, and the method to generations of computational physicists. The reason it lives here is doctrine: we test tools against cases with known answers before trusting them with unknowns. That habit transfers directly to the commercial work, where "the integrator" is a payroll sync and "the known answer" is a hand-reconciled month.