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 $M= S^n$, 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.