Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Evgeny A. Mityushov

TL;DR

Problem: understand rigid body dynamics on $SU(2)$ through a curvature-based organizing principle. Method: decompose the Euler-Poisson system into inertial curvature $K_{ ext{geo}}$ and external curvature $K_{ ext{ext}}$, forming the curvature form $\dot{\Omega} = K_{ ext{geo}}(\Omega) + K_{ ext{ext}}(\Gamma)$, and classify regimes by curvature balance to reveal a curvature-balanced $(2,2,1)$ regime that yields pure precession. Contributions: a unified curvature atlas linking classical integrable tops to curvature regimes, explicit pure-precession construction for $(2,2,1)$, a curvature-based control framework GCCT, and a schematic curvature diagram. Significance: provides a geometric organizing principle for left-invariant dynamics on Lie groups and suggests smooth curvature-driven control on $S^3$ with potential extensions to other Lie groups.

Abstract

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.

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

TL;DR

Problem: understand rigid body dynamics on through a curvature-based organizing principle. Method: decompose the Euler-Poisson system into inertial curvature and external curvature , forming the curvature form , and classify regimes by curvature balance to reveal a curvature-balanced regime that yields pure precession. Contributions: a unified curvature atlas linking classical integrable tops to curvature regimes, explicit pure-precession construction for , a curvature-based control framework GCCT, and a schematic curvature diagram. Significance: provides a geometric organizing principle for left-invariant dynamics on Lie groups and suggests smooth curvature-driven control on with potential extensions to other Lie groups.

Abstract

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.

Paper Structure

This paper contains 20 sections, 1 theorem, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Geodesic Dynamics on $SU(2)$
  3. Left--Invariant Metric and Curvature Field
  4. A1. Spherical Metric
  5. A2. Lagrange Metric
  6. A3. Generic Anisotropic Metric
  7. A4. Kovalevskaya Metric
  8. A5. Goryachev--Chaplygin Metric
  9. A6. The Curvature--Balanced Regime $(2,2,1)$
  10. A7. Summary of Curvature Regimes
  11. Heavy Rigid Body Dynamics: Curvature Formulation
  12. B1. Curvature Decomposition of the Euler--Poisson Equations
  13. B2. Lagrange Top as Orthogonally Balanced Curvature
  14. B3. Curvature--Balanced Regime $(2,2,1)$ and Pure Precession
  15. Geometric Applications and Curvature--Driven Control
  16. ...and 5 more sections

Key Result

Proposition 3.1

Let the inertia ratio satisfy $I_1:I_2:I_3 = 2:2:1$ and let $\mu = (1,0,0)$. Then the Euler--Poisson system admits nontrivial solutions with $\Omega(t) \equiv \Omega_0$. They are exactly the initial conditions satisfying Along these trajectories $\Gamma(t)$ undergoes uniform precession around the fixed axis $\Omega_0$.

Figures (1)

  • Figure 1: Schematic curvature atlas in the $(I_2/I_1,I_3/I_1)$--plane.

Theorems & Definitions (4)

  • Remark : Geometric meaning of $K_{\mathrm{geo}}$
  • Proposition 3.1
  • proof : Proof sketch
  • Remark