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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.16592 |
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| _version_ | 1866915684690165760 |
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| author | Mityushov, Evgeny A. |
| author_facet | Mityushov, Evgeny A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16592 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Curvature atlas II: geometric classification of integrable rigid-body regimes Mityushov, Evgeny A. Exactly Solvable and Integrable Systems 2020 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. |
| title | Curvature atlas II: geometric classification of integrable rigid-body regimes |
| topic | Exactly Solvable and Integrable Systems 2020 |
| url | https://arxiv.org/abs/2512.16592 |