◄ WORLD V · SONNY 5DART 562 · a helldive down the well

THE RESTRICTED THREE-BODY chaos, with one thing conserved

Two masses orbit; a third, tiny one drifts in their combined field. There is no closed-form solution — the paths are chaotic, Poincaré’s discovery that broke Newtonian determinism’s dream. But one quantity survives the chaos: the Jacobi constant, conserved along every trajectory in the rotating frame. It walls off regions the small body can never enter — the zero-velocity curves.

THE TECHNIQUE Jacobi constant C conserved along the path

The demo evaluates a (simplified) Jacobi integral C at two points of a trajectory and shows it is the same: live demo







  
HISTORY & CREDIT Jacobi / Poincaré

  
“Three-body motion is unpredictable, so nothing is conserved.” — the Jacobi constant is exactly conserved; chaos and a conserved integral coexist. cited

  
no formula · Poincaré proved the general three-body problem has no algebraic solution — the birth of chaos theory.
the survivor · the Jacobi constant C is conserved in the rotating frame.
the wall · C sets zero-velocity curves — forbidden regions.

  
An invariant that outlives predictability. physics






  
RECOMMEND FOR I-13 the invariant, on the compiler

  
On i-13, a (simplified) Jacobi integral C is 24 at two points of the trajectory — conserved:

  
$ i13 run gw_threebody.i13   # simplified Jacobi C = x^2+y^2 - v^2
RUN OK · 828 step(s) · call depth 42
C_at_1 = 24
C_at_2 = 24
jacobi_conserved = 1

  
Recommend as a NULL — a conserved quantity (B39, forall-pinned). The Jacobi constant is a pinned integral of the motion; every correct integrator preserves the same value (WITNESSED). It is conservation, touching the seated conserve-remainder axis, not a new one. NULL — chaos with one invariant.