HAMILTONIAN AND GRADIENT STRUCTURES IN THE TODA FLOWS

In this paper we consider gradient structures in the dynamics and geom etry of the asymmetric nonperiodic tridiagonal and full Toda flow equa tions. We compare and contrast a number of formulations of the nonperi odic Toda equations. In the case of the full Kostant (asymmetric) Toda flow we explain the role of noncommutative integrability in its quali tative behavior. We describe the relationship between the asymmetric T oda flows and the symmetric and indefinite Toda flows, and prove in pa rticular that one may conjugate from the full Kostant Toda flows to th e full symmetric Toda flows via a Poisson map. (C) 1998 Elsevier Scien ce B.V.