The Three-Body Problem as Hamiltonian Turbulence: An Isomorphism with 2D Hydrodynamics and the Cascade Cycle

We establish a rigorous isomorphism between the planar three-body problem with zero angular momentum and the dynamics of an ideal incompressible fluid on a surface of constant negative curvature. The configuration space of the three-bodyproblem, equipped with the Jacobi metric and factored by rotations, is conformally equivalent to the Poincar´e upper half-plane H2. The geodesic flow on this space is equivalent to a stationary solution of the 2D Euler equations. We prove that the three-body system possesses a gauge enstrophy invariant Ecal, defined through the curvature of the Jacobi metric, which is exactly conserved. The simultaneousconservation of energy and gauge enstrophy forces an inverse cascade in the space of orbital frequencies—energy is transferred from fast orbital motions to slow secular variations. We define the KAM winding index K as the ratio of the gauge enstrophy to the kinetic energy, weighted by the distance to the nearest rational resonance, and prove that a critical value Kcrit = ρc ≈ 28 marks the topological limit of torus stability. When K exceeds this threshold, invariant KAM tori are destroyed via homoclinic intersection, and the system transitions to chaotic dynamics. After the chaotic episode, the system spontaneously re-organises into a new hierarchical configuration, and the cycle repeats. No dissipation is required—the entire cycle is driven by the geometry of the configuration space. The theory unifies the three-body problem with the turbulence paradigm developed in our previous works andprovides a rigorous, first-principles derivation of the stability boundary, the lifetime distribution, and the recurrence properties of three-body systems.