Three-Body Figure-Eight Choreography

Three equal masses interact under Newtonian gravity in 2D. At the Chenciner-Montgomery initial condition (2000), all three masses chase one another along a single closed figure-eight curve with period T ≈ 6.326. The velocity-Verlet integrator from shared/js/engine/symplectic.js preserves total energy to a part-in-a-million bound and total linear and angular momentum to machine precision. The "dv" slider perturbs the initial velocity of body 3 by a small amount; at dv = 0 the choreography is stable, and at dv = 0.01 the system slowly drifts off the closed curve.

Figure 1. Chenciner-Montgomery figure-eight choreography of three equal point masses under Newtonian gravity in 2D (G = m_i = 1). Method: velocity-Verlet on the 6-DOF Hamiltonian from shared/js/engine/symplectic.js with pairwise gravitational acceleration.

orbit

dv 0.0000

WHAT TO TRY

Three equal masses chase each other down the same figure-eight track. The bottom plot shows the three separations cycling in lockstep, dipping at each close passage; that periodic rhythm is what makes this a choreography.
Nudge the perturbation dv by a hair: the delicate orbit no longer closes and the bodies drift apart. These choreographies live on a knife edge of initial conditions.
Switch orbits from the menu to other known solutions; each has its own separation signature, and energy and angular momentum stay conserved (watch the rail).