Gyrostatic Suslov Problem

In this paper, we investigate the gyrostat under influence of an external potential force with the Suslov nonholonomic constraint: the projection of the total angular velocity onto a vector fixed in the body vanishes. We investigate cases of free gyrostat, the heavy gyrostat in the constant gravity field, and we discuss certain properties for general potential forces. In all these cases, the system has two first integrals: the energy and the geometric first integral. For its integrability, either two additional first integrals or one additional first integral and an invariant $n$-form are necessary. For the free gyrostat we identify three cases integrable in the Jacobi sense. In the case of heavy gyrostat three cases with one additional first integral are identified. Among them, one case is integrable and the non-integrability of the remaining cases is proved by means of the differential Galois methods. Moreover, for a distinguished case of the heavy gyrostat a co-dimension one invariant subspace is identified. It was shown that the system restricted to this subspace is super-integrable, and solvable in elliptic functions. For the gyrostat in general potential force field conditions of the existence of an invariant $n$-form defined by a special form of the Jacobi last multiplier are derived. The class of potentials satisfying them is identified, and then the system restricted to the corresponding invariant subspace of co-dimension one appears to be integrable in the Jacobi sense.