The figure-eight choreographic orbit is a remarkable exact periodic solution of the planar equal-mass three-body problem under Newtonian gravity: three identical bodies chase one another along a single closed curve with time-shifted symmetry. Its existence shows that gravitationally bound motion need not reduce to simple Kepler ellipses when N ≥ 3. Stability is delicate—small perturbations or unequal masses generally destroy the clean periodic pattern—so the orbit is a mathematical showcase more than a generic solar-system trajectory. The simulator illustrates equal-time spacing, symmetric looping, and conservation of energy and angular momentum for this special choreography while using point masses and neglecting relativistic corrections, tidal deformation, and collisions. Students use it as a bridge from textbook two-body Kepler problems to the qualitative chaos and richness of few-body dynamics.
Who it's for: Advanced undergraduate or graduate classical mechanics students studying the three-body problem, periodic orbits, and numerical exploration of chaos.
Key terms
Three-body problem
Choreographic orbit
Figure-eight solution
Periodic orbit
Newtonian gravity
Equal masses
Chaos
Conserved quantities
How it works
The famous Chenciner–Montgomery choreography: three equal masses follow one closed curve in the plane, shifted by 120°. Here G = m = 1 in model units; RK4 integration preserves the orbit for many periods if the step is small enough.
Frequently asked questions
Is the figure-eight orbit stable in nature?
It is only marginally stable in restricted settings; generic perturbations or unequal masses tend to break the symmetry. It is primarily a mathematical landmark and numerical demo.
Why three equal masses?
The symmetric choreography relies on exchanging labels of identical bodies. Unequal masses can still admit periodic solutions, but the simple single-loop eight shape is special to the equal-mass case.
Does angular momentum vanish for this orbit?
The classic planar figure-eight family has zero total angular momentum about the center of mass while still exhibiting rich motion—a striking contrast to Kepler ellipses.
How does this relate to chaos in the solar system?
Real solar system dynamics are mostly perturbative around Kepler orbits, but asteroid resonances and close encounters show sensitive dependence. The figure-eight is a structured exception in an otherwise famously non-integrable problem.