Geometric Characterization of Liouville Integrability via a Curvature Atlas for Rigid-Body Dynamics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918257651351552 |
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| author | Mityushov, Evgeny A. |
| author_facet | Mityushov, Evgeny A. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_18625 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric Characterization of Liouville Integrability via a Curvature Atlas for Rigid-Body Dynamics Mityushov, Evgeny A. Exactly Solvable and Integrable Systems 2020 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. |
| title | Geometric Characterization of Liouville Integrability via a Curvature Atlas for Rigid-Body Dynamics |
| topic | Exactly Solvable and Integrable Systems 2020 |
| url | https://arxiv.org/abs/2512.18625 |