Action-angle Maps and Scattering Theory for Some Finite-dimensional Integrable Systems. III. Sutherland Type Systems and their Duals

Abstract

We present an explicit construction of an action-angle map for the nonrelativistic $N$- particle Sutherland system and for two different generalizations thereof, one of which may be viewed as a relativistic version. We use the map to obtain detailed information concerning dynamical issues such as oscillation periods and equilibria, and to obtain simple formulas for partition functions. The nonrelativistic and relativistic Sutherland systems give rise to dual integrable systems with a solitonic long-time asymptotics that is explicitly described. We show that the second generalization is self-dual, and that its reduced phase space can be densely embedded in ${\mathbf{P}}^{N-1}$ with its standard Kähler form, yielding commuting global flows. In a certain limit the reduced action-angle map converges to the quotient of Fourier transformation on ${\mathbf{C}}^N$ under the standard projection ${\mathbf{C}}^N\backslash \{0\}\to {\mathbf{P}}^{N-1}$.