arXiv subjects

Evgeny A. Mityushov

Publications and source records attributed to Evgeny A. Mityushov.

3 recordsLinked to original sources

arXiv2025

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

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.

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arXiv2025

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

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.

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arXiv2025

Curvature-driver d.dynamics on $S^3$: a geometric atlas

We develop a geometric atlas of dynamical regimes on the rotation group SU(2), combining geodesic flows, heavy rigid body dynamics, and a curvature-based decomposition of the Euler-Poisson equations. We represent the equations of motion in a curvature form that interprets rigid-body motion as the interaction of inertial and external curvature fields. This unified viewpoint recovers classical integrable cases (Lagrange, Kovalevskaya, Goryachev-Chaplygin) from a single geometric mechanism and clarifies their geodesic prototypes on SU(2). The central new result is the identification and geometric explanation of a pure-precession family in the inertia ratio (2,2,1), obtained from a curvature-balanced geodesic regime with the same inertia ratio. The corresponding pure-precession regime for the (2,2,1) heavy top was first identified in previous work; here we place it into a curvature-based atlas and interpret it as a balance between inertial and external curvature fields. We also exhibit a schematic curvature diagram organizing the main dynamical regimes. Finally, we outline GCCT (Geometric Curvature Control Theory), a curvature-driven approach to control on S^3 designed to produce smooth globally regular controls suitable for benchmark maneuvers; a detailed comparison with Pontryagin-type optimal solutions is left for future work.

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