Note on free symmetric rigid body motion

Abstract

We consider the Euler equations of motion of a free symmetric rigid body around a fixed point, restricted to the invariant subspace given by the zero values of the corresponding linear Noether integrals. In the case of the SO(n − 2)-symmetry, we show that almost all trajectories are periodic and that the motion can be expressed in terms of elliptic functions. In the case of the SO(n − 3)-symmetry, we prove the solvability of the problem by using a recent Kozlov’s result on the Euler-Jacobi-Lie theorem.