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

Abstract

A natural extension of the free rigid body dynamics to the unitary group U(n) is considered. The dynamics is described by the Euler equation on the Lie algebra u(n), which has a bi-Hamiltonian structure, and it can be reduced onto the adjoint orbits, as in the case of the SO(n). The complete integrability and the stability of the isolated equilibria on the generic orbits are considered by using the method of Bolsinov and Oshemkov. In particular, it is shown that all the isolated equilibria on generic orbits are Lyapunov stable.

Singularities of Integrable Systems and Algebraic Curves

“…We illustrate Theorem 1.1 with two concrete examples. The first one (shift of argument systems, Section 7.1) generalizes recent results of T. Ratiu and D. Tarama [35]. The second example is the Lagrange top (Section 7.2).…”

“…Remark 7.6. The first two statements of Corollary 7.5 have been recently obtained by T. Ratiu and D. Tarama by means of direct computations (see [35] for the first statement and [44] for the second statement). The interest in singularities for shift of argument systems comes from the fact that these systems can be interpreted as integrable geodesic flows on the corresponding Lie groups, or, in V. Arnold's terminology, as generalized rigid bodies.…”

“…As a result, by now there exists quite an accomplished theory that classifies main types of singularities for integrable systems. However, there exist almost no efficient techniques for describing singularities of particular systems, and except for several examples (see, for instance, recent works [8,13,35]), singularities of multidimensional systems have not been understood.…”