Relative equilibria in the unrestricted problem of a sphere and symmetric rigid body

TL;DR

This paper provides a comprehensive geometric method to identify all relative equilibria in a two-body problem involving a sphere and a symmetric rigid body, extending previous results and discovering a new type of precession.

Contribution

It generalizes Kinoshita's work by showing equilibria can be found via two algebraic equations and introduces a new 'conic' precession class with unique motion characteristics.

Findings

Three classes of relative equilibria identified: cylindrical, inclined co-planar, and conic precessions.

Equilibria can be determined by solving at most two algebraic equations.

Discovered a new conic precession where the point-mass and the body's center move in different planes.

Abstract

We consider the unrestricted problem of two mutually attracting rigid bodies, an uniform sphere (or a point mass) and an axially symmetric body. We present a global, geometric approach for finding all relative equilibria (stationary solutions) in this model, which was already studied by Kinoshita (1970). We extend and generalize his results, showing that the equilibria solutions may be found by solving at most two non-linear, algebraic equations, assuming that the potential function of the symmetric rigid body is known explicitly. We demonstrate that there are three classes of the relative equilibria, which we call "cylindrical", "inclined co-planar", and "conic" precessions, respectively. Moreover, we also show that in the case of conic precession, although the relative orbit is circular, the point-mass and the mass center of the body move in different parallel planes. This solution has been yet not known in the literature.