The Toda Lattice, Old and New

Originally a model for wave propagation on the line, the Toda lattice is a wonderful case study in mechanics and symplectic geometry. In Flaschka’s variables, it becomes an evolution given by a Lax pair on the vector space of Jacobi matrices, with a very special asymptotic behavior. It is a completely integrable system on the coadjoint orbit of the upper triangular group, for which the Liouville-Arnold tori are parameterized in terms of norming constants (which play the role of discrete inverse variables in analogy to the solution by inverse scattering of KdV) and the more recent bidiagonal coordinates (which parameterize also non-Jacobi tridiagonal matrices and reduce asymptotic questions to local theory). Larger phase spaces for the Toda lattice lead to the study of isospectral manifolds and different coadjoint orbits. Moreover, the time one map of the associated flow is computed by a familiar algorithm in numerical linear algebra. The text is mostly expositive and quite self contained, presenting alternative formulations of familiar results and a few applications to numerical analysis.