Nahm's Equations and Generalizations of the Neumann System

Abstract

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 = Sn, 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.