Contact magnetic geodesic and sub-Riemannian flows on Vn,2 and integrable cases of a heavy rigid body with a gyrostat

Abstract

We prove the integrability of magnetic geodesic flows of -invariant Riemannian metrics on the rank two Stefel variety with respect to the magnetic field, where is the standard contact form on and is a real parameter.Also, we prove the integrability of magnetic sub-Riemannian geodesic flows for -invariant sub-Riemannian structures on. All statements in the limit imply the integrability of the problems without the influence of the magnetic field. We also consider integrable pendulum-type natural mechanical systems with the kinetic energy defined by -invariant Riemannian metrics. For, using the isomorphism, the obtained integrable magnetic models reduce tointegrable cases of the motion of a heavy rigid body with a gyrostat around a fixed point:the Zhukovskiy – Volterra gyrostat, the Lagrange top with a gyrostat, and the Kowalevskitop with a gyrostat. As a by-product we obtain the Lax presentations for the Lagrangegyrostat and the Kowalevski gyrostat in the fixed reference frame (dual Lax representations).