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Geometric Characterization of Liouville Integrability via a Curvature Atlas for Rigid-Body Dynamics

Evgeny A. Mityushov

TL;DR

The paper establishes a geometric criterion linking Liouville integrability of the heavy top to degeneracy in the inertial curvature field $K_{ ext{geo}}$. By mapping curvature signatures to classical integrable inertia ratios, it unifies the known integrable cases under a curvature atlas and introduces a curvature-deviation functional $oldsymbol{ riangle}(I)$ to quantify proximity to integrability. It further identifies a balanced-mixed $2:2:1$ regime that, while non-integrable, exhibits exact curvature balance and a family of pure-precession motions, and it presents an integrability map in inertia-ratio space to guide classification and control. The framework opens pathways for near-integrable analysis via averaging, extensions to higher-dimensional groups, and curvature-based control approaches (GCCT).

Abstract

We introduce a curvature atlas for left-invariant metrics on SU(2), based on the inertial curvature field derived from the Euler-Poincare equations. We prove that the classical integrable cases of the heavy top--spherical, Lagrange, Kovalevskaya, and Goryachev-Chaplygin--correspond precisely to degenerate curvature signatures of this field, namely isotropic, orthogonally split, and symmetric-pair signatures. This yields a geometric necessary and sufficient condition for Liouville integrability: the geodesic flow (and the heavy top with axis-symmetric potential) is integrable if and only if the curvature signature is degenerate. Beyond the classical list, the atlas reveals a balanced-mixed regime (inertia ratio 2:2:1) that, while non-integrable, admits an exact curvature-balance relation and a family of pure-precession solutions. We formulate a curvature deviation functional quantifying the distance to integrability, describe near-integrable dynamics near the 2:2:1 regime, and present a complete integrability map in the plane of inertia ratios. The work provides a unified geometric framework for classifying, perturbing, and controlling rigid-body systems.

Geometric Characterization of Liouville Integrability via a Curvature Atlas for Rigid-Body Dynamics

TL;DR

The paper establishes a geometric criterion linking Liouville integrability of the heavy top to degeneracy in the inertial curvature field . By mapping curvature signatures to classical integrable inertia ratios, it unifies the known integrable cases under a curvature atlas and introduces a curvature-deviation functional to quantify proximity to integrability. It further identifies a balanced-mixed regime that, while non-integrable, exhibits exact curvature balance and a family of pure-precession motions, and it presents an integrability map in inertia-ratio space to guide classification and control. The framework opens pathways for near-integrable analysis via averaging, extensions to higher-dimensional groups, and curvature-based control approaches (GCCT).

Abstract

We introduce a curvature atlas for left-invariant metrics on SU(2), based on the inertial curvature field derived from the Euler-Poincare equations. We prove that the classical integrable cases of the heavy top--spherical, Lagrange, Kovalevskaya, and Goryachev-Chaplygin--correspond precisely to degenerate curvature signatures of this field, namely isotropic, orthogonally split, and symmetric-pair signatures. This yields a geometric necessary and sufficient condition for Liouville integrability: the geodesic flow (and the heavy top with axis-symmetric potential) is integrable if and only if the curvature signature is degenerate. Beyond the classical list, the atlas reveals a balanced-mixed regime (inertia ratio 2:2:1) that, while non-integrable, admits an exact curvature-balance relation and a family of pure-precession solutions. We formulate a curvature deviation functional quantifying the distance to integrability, describe near-integrable dynamics near the 2:2:1 regime, and present a complete integrability map in the plane of inertia ratios. The work provides a unified geometric framework for classifying, perturbing, and controlling rigid-body systems.

Paper Structure

This paper contains 7 sections, 4 theorems, 11 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Curvature Field and Its Signatures
  3. Curvature Degeneracy and Liouville Integrability: Main Theorem
  4. The Balanced-Mixed Regime $2:2:1$
  5. Curvature Deviation and Near-Integrable Dynamics
  6. Integrability Map and Applications
  7. Discussion and Outlook

Key Result

Theorem 3.1

Let $I=\operatorname{diag}(I_{1},I_{2},I_{3})$ define a left-invariant metric on $SU(2)$ and let $K_{\text{geo}}$ be given by (1). The following three statements are equivalent:

Figures (1)

  • Figure 1: Integrability map in the $(I_{2}/I_{1},I_{3}/I_{1})$-plane. The shaded region $y\leq x$ corresponds to physically admissible inertia ratios after reordering axes. Thick curves mark the classical integrable cases; the orange square indicates the balanced-mixed regime ($2:2:1$). Dashed circles schematically represent level sets of the curvature deviation $\Delta(I)$.

Theorems & Definitions (10)

  • Definition 2.1: Curvature signature
  • Theorem 3.1: Curvature-Integrability Correspondence
  • proof
  • Remark 3.2
  • Theorem 4.1: Curvature Balance without Integrability
  • proof
  • Definition 5.1: Curvature deviation
  • Proposition 5.2: Properties of $\Delta$
  • Theorem 5.3: Near-integrable dynamics
  • proof : Sketch of proof