D01 — Figure-Eight 3-Body Choreography
| Field | Value |
|---|---|
| Domain | Dynamical Systems |
| System | Equal-mass 3-body gravitational problem in ℝ² |
| Group | D₆ (dihedral symmetry of the figure-eight) |
| H^k tier | H⁰ |
| ISA | Origami (β → ∞) |
| Status | Validated |
| Opcodes | ORBIT · LABEL |
| Paper | Paper 552 |
Physical system
Three equal masses chase each other around a figure-eight curve in the plane, discovered numerically by Moore (1993) and proved to exist by Chenciner & Montgomery (2000). The orbit is periodic with period T ≈ 6.3259 (in units where G = m = 1) and is the unique (up to symmetry) stable choreography of three equal masses in ℝ².
Target category
Symp — the category of symplectic manifolds and canonical maps. The phase space is T*ℝ⁶ (positions and momenta of 3 bodies); the choreography is a closed orbit in the zero-angular-momentum, zero-centre-of-mass submanifold.
Interpretation functor
F: C → Symp defined by:
| Opcode | F(opcode) |
|---|---|
| ORBIT | Closed periodic trajectory in shape space; eigenvalue = winding number = 1 |
| LABEL | Symmetry label under D₆: the figure-eight has a unique choreography class |
ISA programme
INIT: LABEL[q₁, q₂, q₃ | figure-eight IC] -- Chenciner-Montgomery initial conditions
FLOW: ORBIT[Φ_t : T*ℝ⁶ → T*ℝ⁶] -- Hamiltonian flow for time T
CLOSE: ORBIT[q(T) = q(0)] -- check closure (tropical fixed point)
PERIOD: LABEL[T ≈ 6.3259] -- period eigenvalue
Computable output
- Period T ≈ 6.3259 (Chenciner-Montgomery, confirmed to 10 significant figures numerically).
- Winding number = 1: the ORBIT closes after exactly one traversal — this is the definition of an H⁰ tropical fixed point. The figure-eight is the unique minimum of the action functional on the space of choreographies with D₆ symmetry.
- H⁰ interpretation: in the ISA, a closed orbit is a tropical fixed point — the β → ∞ limit of a Gibbs distribution over trajectories. The figure-eight is the ground state (lowest action) of 3-body choreography space. All other choreographies require H¹ TWIST (non-trivial topology) or H² BIND (non-abelian holonomy in higher dimensions).
Validation
- Chenciner & Montgomery (2000) proved existence via variational minimisation of the action on the D₆-symmetric path space — exactly the ORBIT fixed-point condition.
- Numerically stable: small perturbations return to the orbit (local minimum of action), confirming it as a β → ∞ attractor.
- Connection to Paper 552: figure-eight = H⁰ entry point of the choreography ISA ladder. H¹ entries include the Lagrange equilateral triangle (TWIST phase) and H² = G₂ choreography in ℝ⁷ (see D02).