Lagrangian relative equilibria for a gyrostat in the three-body problem: bifurcations and stability

Abstract

In this paper we consider the non-canonical Hamiltonian dynamics of a gyrostat in the frame of the three-body problem. Using geometric/mechanic methods we study the approximate dynamics of the truncated Legendre series representation of the potential of an arbitrary order. Working in the reduced problem, we study the existence of relative equilibria that we refer to as Lagrange type following the analogy with the standard techniques. We provide necessary and sufficient conditions for the linear stability of Lagrangian relative equilibria if the gyrostat morphology form is close to a sphere. Thus, we generalize the classical results on equilibria of the three-body problem and many results on them obtained by the classic approach for the case of rigid bodies.

Periodic solutions induced by an upright position of small oscillations of a sleeping symmetrical gyrostat

“…These rotors may rotate with respect to the platform in such a way that the mass distribution within the system as a whole is not altered; this will produce an internal angular momentum, designated as gyrostat momentum, which will be normally regarded as a constant. Note that when this constant vector is zero, the motion of the system is reduced to the motion of a rigid solid, see for instance Figure 1 where a gyrostat in the frame of the three body problem is presented, and [3,4] or [5] for more details on this class of mechanical systems. The objective of this paper is to provide, using the averaging theory, a system of nonlinear equations whose simple zeros provide periodic solutions of the differential system (1).…”

Stabilizing the coupled orbit–attitude dynamics of a rigid body in a J2 gravity field using Hamiltonian structure

“…When the gravitationally coupled orbit-attitude dynamics model of a rigid body in a J 2 gravity field is adopted in the future asteroid mission design, the relative equilibrium of the rigid body can be used as the nominal motion, which has also been shown by other studies in the three body problem [14,4,5]. Then, the stabilization of relative equilibria is necessary during the mission, since the stability of relative equilibria of uncontrolled system cannot be always guaranteed by the parameters of spacecraft.…”