Newtonian Equations of the Toda Type

Abstract

Actually,the integrable system originally discovered by Toda was (3.1.1) — a lattice of particles interacting with nearest neighbors via forces exponentially depending on distances. Only later was it rewritten in the Flaschka variables (a,b), i.e., in the form (3.1.3). However, it turns out that the system (3.1.3) has a much richer structure than (3.1.1). In particular, it is a tri-Hamiltonian system. It turns out that each one of invariant Poisson structures, and some of their linear combinations, allow parametrizations by means of canonically conjugate variables (x,p)which leads to a whole variety of Newtonian equations of motions hidden in (3.1.3). The present chapter is devoted to elaborating the relevant systems, along with their discretizations. Here we list the main Newtonian equations and their discrete time counterparts arising from (3.1.3) and (3.8.2), respectively, via different parametrizations of the (a,b) variables by canonically conjugate ones.