A Hamiltonian Study Of The Stability and Bifurcations For The Satellite Problem
M. C. Muñoz-Lecanda, Miguel Rodriguez-Olmos, Miguel Teixidó-Román
TL;DR
This work analyzes the satellite problem as a Hamiltonian system with symmetry, focusing on how attitude-orbit coupling shapes relative equilibria. Building on geometric methods, the authors derive and analyze a second-order model with $H_2=K+V_2$ and $V_2(R)=-\frac{1}{R}-\frac{1}{2R^3}+\frac{3}{2R^5}\mathbf{R}\cdot \mathbf{I}\mathbf{R}$, applying the Reduced Energy Momentum method to obtain existence, stability, and bifurcation results for both asymmetric and axisymmetric bodies. A key contribution is the analytic description of conical equilibria in the axisymmetric case, shown to persist for all orbital radii, along with comprehensive stability conditions and bifurcation diagrams that parallel, yet extend beyond, the restricted problem. The findings illuminate the rich organization of the coupled orbital-attitude dynamics in low-orbit regimes and provide rigorous criteria for when qualitatively different motions emerge or disappear. Overall, the paper delivers a rigorous, symmetry-driven framework for predicting stability and bifurcations in the satellite problem with finite-body effects.
Abstract
We study the dynamics of a rigid body in a central gravitational field modeled as a Hamiltonian system with continuous rotational symmetries following the geometrical framework of Wang et al. Novelties of our work are the use the Reduced Energy Momentum for the stability analysis and the treatment of axisymmetric bodies. We explicitly show the existence of new relative equilibria and study their stability and bifurcation patterns.