Symmetry properties and bi-Hamiltonian structure of the Toda lattice

Abstract

We carry out a systematic analysis of the Toda lattice equations developing a method which extends the symmetry approach formalism to discrete one-dimensional systems. We find a hereditary operator which admits a symplectic-implectic factorization. As a consequence of this property, we derive the Hamiltonian and the bi-Hamiltonian structure, together with the constants of motion and a set of infinitely-many commuting Lie-Bäcklund symmetries of the Toda chain.