Complete involutive algebras of functions on cotangent bundles of homogeneous spaces

Abstract

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.

Magnetic geodesic flows on coadjoint orbits

“…is a complete commutative subset of R [v] Ga with respect to the canonical brackets {·, ·} 0 v (see [4,7]). Here R [g] G is the algebra of Ad G -invariant polynomials on g. Using the method of [2], it can be verified that B a is a complete commutative set with respect to the magnetic Poisson bracket (18) …”

“…Note that δ does not depend on ǫ: for a generic η ∈ v we have equality dim O(η + ǫa) = dim O(η) for all ǫ ∈ R (see [4,7]). Therefore, the influence of the magnetic fields ǫΩ, ǫ ∈ R reflects as a deformation of the foliation of the phase space T * O(a) by invariant tori.…”

“…As the magnetic field increases, the magnetic geodesic lines become more curved. Here we assume that the distribution D is bracket generating (see [10,4] for more details).…”

Bi-Poisson Structures and Integrability of Geodesic Flow on Homogeneous Spaces

“…The analogous result for the unitary group G = U(n) was obtained by Bolsinov and Jovanovic in their paper [BJ3,Theorem 3.4]. The proof of their theorem is based on a verification of some sufficient conditions using canonical matrix representations of semisimple elements of the Lie algebra u(n) and, as remarked in [BJ3], may be generalized for the case of compact classical Lie groups G = SO(n) or Sp(n).…”

A variational problem on Stiefel manifolds

“…Theorem 4 of Moser and Veselov (1991) gives a set of isospectral deformations for the discrete geodesic flow, which can be viewed as a discrete analogue of the parameterdependent Lax representation. Recent results by Bolsinov and Jovanovic (2004) We present a generalization of the Lax pair form for the equations of motion and show how this reduces to the classical Lax pair form in the case of the rigid body equations and the geodesic flow on the ellipsoid. The integrability of the rigid body equations by a Lax pair formulation with parameter had been shown by Manakov (1976); Mischenko and Fomenko (1978) showed that a similar formalism exists for any semisimple Lie group.…”