Gyroscopic Chaplygin systems and integrable magnetic flows on spheres

We introduce and study the Chaplygin systems with gyroscopic forces with a special emphases on the important subclass of such systems with magnetic forces. This class of nonholonomic systems is natural and has not been treated before. We describe the gyroscopic Chaplygin systems on fiber spaces and the reduction procedure for the corresponding $G$--Chaplygin systems. The existence of an invariant measure and the problem of Hamiltonization are studied, both within the Lagrangian and the almost-Hamiltonian framework. In addition, we introduce problems of rolling of a ball with the gyroscope without slipping and twisting over a plane and a sphere in $\mathbb R^n$ as examples of gyroscopic $SO(n)$--Chaplygin systems. We describe an invariant measure and provide a class of examples, which allow the Chaplygin Hamiltonization, and prove the integrability of the obtained magnetic geodesic flows on the sphere $S^{n-1}$. In particular, we introduce the generalized Demchenko case in $\mathbb R^n$, when the inertia operator of the system (ball + gyroscope) is proportional to the identity operator. The reduced system is automatically Hamiltonian and represents the magnetic geodesic flow on the sphere $S^{n-1}$ endowed with the round-sphere metric, under the influence of the homogeneous magnetic field. The magnetic geodesic flow problem on the two-dimensional sphere is well known, but for $n>3$ was not studied before. We perform explicit integrations in elliptic functions of the systems for $n=3$ and $n=4$, and provide the case study of the solutions in both dimensions.