Stability of Equilibria for the $\mathfrak{so}(4)$ Free Rigid Body

Abstract

The stability for all generic equilibria of the Lie-Poisson dynamics of the so(4) rigid body dynamics is completely determined. It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of so(n) are equilibrium points for the rigid body dynamics. In the case of so(4) there are three coordinate type Cartan subalgebras whose intersection with a regular adjoint orbit gives three Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body in Communicated by A. Bloch. P. Birtea · I. Caşu ( )

The stability problem and special solutions for the 5-components Maxwell–Bloch equations

“…where the second commuting constant of motion is I| Oc . We will study the stability problem of equilibria on regular leaves O c analogously to the approach used in [2]. The equilibria of the Hamiltonian system (2.1) can be divided in two types:…”

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].…”

“…A wellknown result for usual free rigid body dynamics in R 3 , which is a Hamiltonian system whose configuration space is the Lie group SO (3), states that rotations about the long and short principal axes are Lyapunov stable, whereas rotations about the middle principal axis are unstable. 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].…”

Stability of relative equilibria of multidimensional rigid body

“…Note that there is a case with a tangency point in the upper half-plane ( Figure 6) when Theorem 1 is not applicable. It is claimed in [3,7] that this rotation is unstable, however this conclusion seems to be incorrect. This follows from the results of [24] and can also be deduced from the bifurcation diagrams constructed by Oshemkov [2].…”