Geometry and integrability of Euler-Poincaré-Suslovequations

Abstract

We consider non-holonomic geodesic flows of left-invariant metrics andleft-invariant non-integrable distributions on compact connected Liegroups. The equations of geodesic flows are reduced to theEuler-Poincaré-Suslov equations on the corresponding Lie algebras.The Poisson and symplectic structures give rise tovarious algebraic constructions of the integrableHamiltonian systems. On the other hand, non-holonomicsystems are not Hamiltonian and the integration methodsfor non-holonomic systems are much less developed. Inthis paper, using chains of subalgebras, we giveconstructions that lead to a large set of first integralsand to integrable cases of the Euler-Poincaré-Suslovequations. Furthermore, we give examples of non-holonomicgeodesic flows that can be seen as a restriction ofintegrable sub-Riemannian geodesic flows.