We consider a class of dynamical systems on a compact Lie group G with a left-invariant metric and right-invariant nonholonomic constraints (so called LR systems) and show that, under a generic condition on the constraints, such systems can be regarded as generalized Chaplygin systems on the principle bundle G → Q = G/H, H being a Lie subgroup. In contrast to generic Chaplygin systems, the reductions of our LR systems onto the homogeneous space Q always possess an invariant measure.We study the case G = SO(n), when LR systems are multidimensional generalizations of the Veselova problem of a nonholonomic rigid body motion which admit a reduction to systems with an invariant measure on the (co)tangent bundle of Stiefel varieties V (k, n) as the corresponding homogeneous spaces.For k = 1 and a special choice of the left-invariant metric on SO(n), we prove that after a time substitution, the reduced system becomes an integrable Hamiltonian system describing a geodesic flow on the unit sphere S n−1 . This provides a first example of a nonholonomic system with more than two degrees of freedom for which the celebrated Chaplygin reducibility theorem is applicable for any dimension. In this case we also explicitly reconstruct the motion on the group SO(n).

Under periodic boundary condition in the transverse direction, we calculate the averaged zero-temperature two-terminal conductance (< G >) and its statistical fluctuations (< (δG) 2n > for n ≤ 4) at the critical point of integer quantum Hall plateau transitions. We find universal values for < G >= (0.58 ± 0.03) e 2 h , and < (δG) 2n >= (

Homogeneous spaces of all compact Lie groups admit Riemannian metrics with completely integrable geodesic flows by means of C ∞ -smooth integrals [9,10]. The purpose of this paper is to give some constructions of complete involutive algebras of analytic functions, polynomial in velocities, on the (co)tangent bundles of homogeneous spaces of compact Lie groups. This allows us to obtain new integrable Riemannian and sub-Riemannian geodesic flows on various homogeneous spaces, such as Stiefel manifolds, flag manifolds and orbits of the adjoint actions of compact Lie groups.

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The paper studies a natural n-dimensional generalization of the classical nonholonomic Chaplygin sphere problem. We prove that for a specific choice of the inertia operator, the restriction of the generalized problem onto zero value of the SO(n-1)-momentum mapping becomes an integrable Hamiltonian system after an appropriate time reparametrization.

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Abstract. This is a survey on finite-dimensional integrable dynamical systems related to Hamiltonian G-actions. Within a framework of noncommutative integrability we study integrability of G-invariant systems, collective motions and reduced integrability. We also consider reductions of the Hamiltonian flows restricted to their invariant submanifolds generalizing classical Hess-Appel'rot case of a heavy rigid body motion.