Tri-Hamiltonian Toda Lattice and a Canonical Bracket for Closed Discrete Curves

Abstract

Flows on (or variations of) discrete curves in $\mathbb{R}^2$ give rise to flows on a subalgebra of functions on that curve. For a special choice of flows and a certain subalgebra this is described by the Toda lattice hierarchy. Here it is shown that the canonical symplectic structure on $\mathbb{R}^{2N}$, which can be interpreted as the phase space of closed discrete curves in $\mathbb{R}^2$ with length N, induces Poisson commutation relations on the above-mentioned subalgebra which yield the tri-Hamiltonian poisson structure of the Toda lattice hierarchy.