Perturbations of the Hess-Appelrot and Lagrange cases in the rigid body dynamics

The Lagrange case in the rigid body dynamics is completely integrable, with a family of invariant tori supporting periodic or quasi-periodic motion. We study perturbations of this case. In the non-periodic case the KAM theory predicts no changes in the evolution. In the periodic cases one expects existence of isolated limit cycles, which can be studied using Melnikov functions. We find these cycles in the case when the invariant torus is close to so-called critical circle. The presented approach is analogous to our previous analysis of the Hess–Appelrot case. In particular, we show that the number of created limit cycles in the latter case is uniformly bounded.