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.我们建立了零角动量平面三体问题（planar three-body problem）与常负曲率曲面上理想不可压缩流体（ideal incompressible fluid）动力学之间的严格同构关系。配备雅可比度量（Jacobi metric）并经旋转因子约化后的三体问题位形空间（configuration space），共形等价于庞加莱上半平面（Poincaré upper half-plane）H²。该空间上的测地流（geodesic flow）等价于二维欧拉方程（2D Euler equations）的定常解。我们证明，三体系统存在由雅可比度量曲率定义的规范涡量不变量（gauge enstrophy invariant）E_cal，且该不变量严格守恒。能量与规范涡量的同时守恒，会导致轨道频率空间中出现逆级串（inverse cascade）——能量将从快速轨道运动传递至缓慢的久期变化（secular variations）。我们将KAM环绕指数（KAM winding index）K定义为规范涡量与动能（kinetic energy）的比值，并以到最近有理共振（rational resonance）的距离作为权重；同时证明临界值K_crit=ρ_c≈28标志着环面稳定性的拓扑极限。当K超过该临界阈值时，不变KAM环面（KAM tori）将通过同宿相交（homoclinic intersection）被破坏，系统随之转变为混沌动力学（chaotic dynamics）行为。混沌阶段结束后，系统会自发重组为新的层级结构构型（hierarchical configuration），随后循环往复。整个过程无需耗散机制——循环完全由位形空间的几何性质所驱动。本理论将三体问题与我们此前研究中提出的湍流范式（turbulence paradigm）相统一，并为三体系统的稳定性边界、寿命分布以及复发特性提供了严格的第一性原理（first-principles）推导。