Nahm's Equations and Generalizations of the Neumann System

In this paper a completely integrable Hamiltonian system on T * M is constructed for every Riemannian symmetric space M . We show that the solutions of this system correspond to the solutions of Nahm's equations for suitably chosen maps. Nahm's equations were introduced by Nahm as a rewriting of Bogomolny equations for magnetic monopoles. We represent our system as a degeneration of a certain case of Hitchin's algebraically integrable system. We prove the complete integrability of our system by means of this representation. Some concrete examples of our Hamiltonian system on T * M are described. When M = S n , we obtain the classical system of C. Neumann. If the configuration space M of our system is the n ‐dimensional hyperbolic space, we get the Minkowskian analogue of the C. Neumann system. Other examples that we describe are a many‐body C. Neumann system, a spherical pendulum, and a spherical pendulum with an additional magnetic force. 1991 Mathematics Subject Classification : 11D25, 11G05, 14G05.