Tri-hamiltonian Toda lattice and a canonical bracket for closed discrete curves

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 hierachy. In the paper 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 hierachy.