Gyroscopic Chaplygin systems and integrable magnetic flows on spheres

We introduce and study the Chaplygin systems with gyroscopic forces. This natural class of nonholonomic systems has not been treated before. We put a special emphasis on the important subclass of such systems with magnetic forces. 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 over a sphere in $\mathbb R^n$ as examples of gyroscopic $SO(n)$--Chaplygin systems. We describe an invariant measure and provide examples of $SO(n-2)$--symmetric systems (ball with gyroscope) that allow the Chaplygin Hamiltonization. In the case of additional $SO(2)$--symmetry we prove that the obtained magnetic geodesic flows on the sphere $S^{n-1}$ are integrable. In particular, we introduce the generalized Demchenko case in $\mathbb R^n$, where the inertia operator of the system is proportional to the identity operator. The reduced systems are automatically Hamiltonian and represent the magnetic geodesic flows on the spheres $S^{n-1}$ endowed with the round-sphere metric, under the influence of a 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 situations.