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

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.