Integrable 2D Time-Irreversible Systems with a Cubic Second Integral

Abstract EN

We construct a very rare integrable 2D mechanical system which admits a complementary integral of motion cubic in the velocities in the presence of conservative potential and velocity-dependent (gyroscopic) forces.

Special cases are given interpretation as a motion of a particle on a sphere endowed with a Riemannian metric, a particle in the Euclidean plane, and new generalizations of two cases of motion of a rigid body with a cubic integral, known by names of Goriachev-Chaplygin and Goriachev.