The construction of Poincaré-Chetayevand Smale bifurcation diagrams for conservative non-holonomic systems with symmetry

The problem of the existence of first integrals which are linear functions of the generalized velocities (momenta and quasi-velocities) is discussed for conservative non-holonomic Chaplygin systems with symmetry, as well as methods for investigating the existence, stability, and bifurcation of the steady motions of such systems. These methods are based on the classical methods of Routh-Salvadori, Poincaré-Chetayev, and Smale, but unlike the latter they do not require a knowledge of the explicit form of the linear integrals. The general conclusions are illustrated by the example of the problem of an ellipsoid of revolution moving on an absolutely rough horizontal surface. It is shown how in this case numerical techniques can be used to construct the Poincaré-Chetayev diagram — a surface in the space of generalized coordinates and constants of linear first integrals corresponding to motions in which the velocities of the non-cyclic coordinates vanish, while those of the cyclic coordinates are constant, and the Smale diagram — a surface in the space of constants of linear first integrals and the energy integral corresponding to these motions.