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

A natural extension of the free rigid body dynamics to the unitary group U(n)U(n) is considered. The dynamics is described by the Euler equation on the Lie algebra u(n)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)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.