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Nahm's Equations and Generalizations of the Neumann System
Author(s) -
Saksida Pavle
Publication year - 1999
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611599001847
Subject(s) - integrable system , mathematics , hamiltonian system , von neumann architecture , pure mathematics , hamiltonian (control theory) , mathematical physics , mathematical analysis , mathematical optimization
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.