Hamiltonian evolutions of twisted polygons in $\mathbb{RP}^n$

Abstract

In this paper we find a discrete moving frame and their associated invariants along projective polygons in RP n , and we use them to describe invariant evolutions of projective N -gons. We then apply a reduction process to obtain a natural Hamiltonian structure on the space of projective invariants for polygons, establishing a close relationship between the projective N -gon invariant evolutions and the Hamiltonian evolutions on the invariants of the flow. We prove that any Hamiltonian evolution is induced on invariants by an invariant evolution of N -gons-what we call a projective realization-and both evolutions are connected explicitly in a very simple way. Finally, we provide a completely integrable evolution (the Boussinesq lattice related to the lattice W 3 -algebra), its projective realization in RP 2 and its Hamiltonian pencil. We generalize both structures to n-dimensions and we prove that they are Poisson, defining explicitly the n-dimensional generalization of the planar evolution (a discretization of the W n -algebra). We prove that the generalization is completely integrable, and we also give its projective realization, which turns out to be very simple.

Darboux transformations and recursion operators for differential-difference equations

“…For nonlinear equation, that is, αβ 0, its Hamiltonian f depends on the coefficients in polynomial P defined by (79). When α 0, we take f = 1 2α ln P(u), otherwise, we take f = u β .…”

“…The Belov-Chaltikian lattice is the Boussinesq lattice related to the lattice W 3 -algebra [78]. In recent paper [79], the authors wrote down the Boussinesq lattice related to the lattice W m -algebra for the dependent variables u 1 , u 2 , · · · , u m and independent variables n and t as follows:…”

Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice

“…Recursively, this means that it also reduces to the N = 3 system, i.e., equation (5.2). Other systems with similar behaviour have been presented in [4].…”

Discrete Moving Frames on Lattice Varieties and Lattice-Based Multispaces

“…We construct the multispace version of the construction to obtain a geometric discretisation. We show that a modification of the construction generates an integrable discretisation which appeared in [47]. The study of how these different discretisations might be related is underway.…”

“…In that paper we prove discrete recursion formulae for small computable generating sets of invariants, which we call the discrete Maurer-Cartan invariants, and investigated their syzygies, that is, their recursion relations. The main application to date has been to discrete integrable systems, with the authors of [47] proving that discrete Hamiltonian structures for W n -algebras can be obtained via a reduction process. We note that a sequence of moving frames was also used in [35] to minimise the accumulation of errors in an invariant numerical method.…”