Curvature atlas II: geometric classification of integrable rigid-body regimes

Abstract

This paper is the second part of a curvature-based program for rigid-body dynamics on SU2. In Part I, Curvature-Driven Dynamics on S3: A Geometric Atlas, we introduced the inertial curvature field Kgeo associated with a left-invariant metric on SU2, constructed a geometric Atlas of curvature regimes, and identified the inertia ratio 2:2:1 as a curvature-balanced regime giving rise to a pure-precession family for the heavy top, building on the prior dynamical discovery of this regime. Here we develop the curvature Atlas into a classification tool for integrable and near-integrable rigid-body regimes. We show that all classical integrable heavy-top cases (Euler, Lagrange, Kovalevskaya, Goryachev-Chaplygin) correspond to specific degenerate curvature signatures of Kgeo. In each case the inertia tensor imposes a simple algebraic structure on the inertial curvature field: vanishing of one component, symmetric curvature pairs, or orthogonal curvature splitting. We formulate and prove a curvature classification theorem describing these signatures and their relation to integrability. We then single out the mixed anisotropic ratio 2:2:1 as a minimally nondegenerate curvature-balanced regime: it destroys algebraic integrability while preserving an exact curvature balance, giving rise to pure precession for the heavy top. Finally, we introduce a curvature deviation functional measuring the distance to the nearest integrable curvature signature, describe near-integrable regimes in a neighbourhood of 2:2:1, and present an integrability map.