Stability of relative equilibria of multidimensional rigid body

Abstract

It is a classical result of Euler that the rotation of a torque-free three-dimensional rigid body about the short or the long axis is stable, whereas the rotation about the middle axis is unstable. This result is generalized to the case of a multidimensional body.

Integrability and Dynamics of the n-Dimensional Symmetric Veselova Top

“…Recall that for a dynamical system, a compact subset S of the phase space is Lyapunov stable if for each neighbourhood U of S there is a neighbourhood V such that any trajectory that enters V lies entirely in U. For the free n-dimensional rigid body, Izosimov [19] has produced a fairly complete analysis of the stability of the relative equilibria. In particular, in Example 2.3 he shows that if J has simple eigenvalues, with J 1 < J 2 < · · · < J n then the 2-dimensional rotations in the principal plane Π i, j are (Lyapunov) stable if and only if |i− j| = 1 (that is, if J i and J j are adjacent in the ordering).…”

Algebraic geometry and stability for integrable systems

“…A weaker version of this result was proved in [18] by means of the bi-Hamiltonian approach. It included an additional requirement that there are no tangency points on the parabolic diagram (i.e.…”

“…Using the algebro-geometric approach, we obtain a multidimensional generalization of this result. We note that this problem has previously been approached by different methods [13][14][15][16][17][18], however no complete solution has been known.…”

The U(n) free rigid body: Integrability and stability analysis of the equilibria

“…The linearization of Hamilton's equations on generic adjoint orbits around the common equilibria is carried out. As opposed to the stability analysis for the SO(n) free rigid body [4,18], these equilibria are all linearly stable. From the linear stability of these equilibria, one can also conclude their Lyapunov stability by using the results in the previous sections and Vey's theorem [35].…”

“…For SO (4), the stability of a certain class of equilibria has been studied in [13] and the complete analysis of the stability for all the equilibria was carried out in [4]. For general SO(n), the stability of a special family has been analyzed in the Ph.D. thesis [33] and, more recently, in [18] which gives the complete analysis of the stability for generic equilibria on the basis of the paper [8].…”