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

Nadja Kutz
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Abstract

Flows on (or variations of) discrete curves in $\R^2$ 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 $\R^{2N}$, which can be interpreted as the phase space of closed discrete curves in $\R^2$ with length $N,$ induces Poisson commutation relations on the above mentioned subalgebra which yield the tri-hamiltonian poisson structure of the Toda lattice hierachy.

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Nadja Kutz. 2003-04-07. Tri-hamiltonian Toda lattice and a canonical bracket for closed discrete curves. https://arxiv.org/abs/math/0304083

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