DOUBLE BRACKET EQUATIONS AND GEODESIC-FLOWS ON SYMMETRICAL SPACES

In this paper we consider the geometry of Hamiltonian flows on the cot angent bundle of coadjoint orbits of compact Lie groups and on symmetr ic spaces. A key idea here is the use of the normal metric to define t he kinetic energy, This leads to Hamiltonian flows of the double brack et type. We analyze the integrability of geodesic flows according to t he method of Thimm. We obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly t ransparent proof of involutivity. We demonstrate for example integrabi lity of the geodesic flow on the real and complex Grassmannians. We al so consider right invariant systems and the generalized rigid body equ ations in this setting.