A Rational Flow for the Periodic Toda Lattice Equations

Dedicated to my long-time friend and colleague, Paul Fuhrmann, on the occasion of his 60 th birthday Abstract We show how a certain class of Hamiltonian systems give rise to diierential equations on spaces of matrices whose elements are rational functions. In particular, we reinterpret the results of Kac and van Moerbeke on the periodic Toda lattice in terms of such diierential equations and relate the action-angle coordinates found by them to the evolution of a certain two by two symmetric matrix of rational functions owing on a space of xed McMillian degree and xed Cauchy index. Realization theory is used to pass from a description of the ow in terms of rational matrices to a description in terms of the original coordinates.