Phase topology of one integrable case of the rigid body motion

Abstract

The reduced system in the Clebsch problem of the motion of a rigid body in fluid treated as the motion of a rigid body about its fixed mass center in a central Newtonian field with zero value of the area integral is a completely integrable natural mechanical system with two degrees of freedom. We find out the phase topology of this system including constructing the bifurcation set of two integrals quadratic with respect to velocities and describing all types of bifurcations of the integral tori. We establish the topology of all singular integral surfaces. Using the contemporary language, one can say that this is the first description of the bifurcations of the types $B$ ($T^2\to 2T^2$) and $C_2$ ($2T^2 \to 2T^2$), for the latter see also arXiv:1408.4548. This investigation includes the description of the foliation into integral surfaces of an invariant neighborhood of a saddle type singularity having two equilibriums on the connected critical integral surface.