Energy Methods in the Stability Problem for the 𝔰𝔬(4) Free Rigid Body

Abstract

For the so(4) free rigid body the stability problem for isolated equilibria has been completely solved using Lie-theoretical and topological arguments. For each case of nonlinear stability previously found we construct a Lyapunov function. These Lyapunov functions are linear combinations of Mishchenko's constants of motion.

Stability of relative equilibria of multidimensional rigid body

“…The solution of the stability problem in dimension four was obtained by Fehér and Marshall [3], and later by another approach by Birtea and Cas ¸u [5], Birtea et al [7]. In this paper these results receive a geometric interpretation (see example 2.3).…”

“…The aim of the present paper is to establish a multidimensional generalization of this result. The problem was studied by a number of authors [2][3][4][5][6][7], however the general answer has only been obtained in dimension four. As the dimension grows, the problem becomes too complicated from the computational point of view when being approached by direct methods.…”

Stability in bihamiltonian systems and multidimensional rigid body

“…Note that parabolic diagrams, which appear naturally as spectral data of the Poisson pencil associated with a rigid body, give a visual interpretation of stability results even in the fourdimensional case, which was studied earlier by direct methods in [13,15,16].…”

“…This problem has been studied by many people, see [12][13][14][15][16][17]. However, no general solution is known.…”

A note on relative equilibria of a free multidimensional rigid body

“…The problem of stability of relative equilibria of a multidimensional rigid body was studied by many authors. See [8,10,12,13,9].…”