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

TL;DR

This work provides a curvature-centric reformulation of rigid-body integrability on SU(2) by using the inertial curvature field $K_{\mathrm{geo}}$ to classify curvature signatures corresponding to classical integrable tops. It establishes a curvature classification theorem that ties spherical, Lagrange, Kovalevskaya, Goryachev-Chaplygin, and the mixed $(2{:}2{:}1)$ inertia ratios to degeneracies in $K_{\mathrm{geo}}$, thereby explaining integrability as a consequence of curvature structure. A curvature-deviation functional $\Delta(I)$ is introduced to measure distance to integrable signatures, enabling a near-integrable analysis and the construction of an integrability map in the plane of inertia ratios $(I_2/I_1,I_3/I_1)$. The $(2{:}2{:}1)$ regime emerges as the first minimally nondegenerate yet curvature-balanced case, predicting slow drift and persistence of near-integrable behavior under small perturbations. Overall, the curvature atlas provides a geometric lens for understanding, predicting, and extending integrable and near-integrable regimes across rigid-body dynamics on Lie groups.

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.

Figures (1)

• Figure 1: Curvature integrability map in the $(I_2/I_1,I_3/I_1)$--plane. The spherical point, Lagrange line, Kovalevskaya and Goryachev--Chaplygin lines, and the mixed $(2{:}2{:}1)$ point are indicated. The curvature deviation $\Delta(I)$ measures the distance to the integrable loci and is small in a neighbourhood of these sets, particularly near $(2{:}2{:}1)$.

Theorems & Definitions (12)

• Definition 2.1: Curvature signature
• Proposition 2.2: Curvature signatures for classical ratios
• proof
• Theorem 3.1: Curvature classification of rigid--body regimes
• Corollary 3.2: Curvature interpretation of classical integrable cases
• Remark
• proof : Proof of Theorem \ref{['thm:curv-class']}
• Definition 4.1: Curvature deviation
• Proposition 4.2
• proof