Nonlinear normal modes for the Toda Chain

The Toda Chain is a nonlinear mass-spring chain which can in principle be integrated analytically by the action-angle theory of classical mechanics. We show that the transformation from physical variables to action variables can be implemented very simply on a computer. We study the correspondence between action variables and the motions of the chain, and find the role of the action variables to be very similar to the role of normal-mode amplitudes in the harmonic chain. This similarity leads to a partly rigorous, partly heuristic normal-mode analysis for the Toda chain. New results are obtained when this normal-mode analysis is applied to certain perturbations of the Toda chain such as the Fermi-Pasta-Ulam chain or the Toda chain with a mass impurity.