Hamiltonian evolutions of twisted gons in $\RP^n$

TL;DR

This paper develops a Hamiltonian framework for the evolution of twisted polygons in projective space, establishing integrability and explicit invariants, with applications to higher-dimensional generalizations and connections to algebraic structures like the W-algebra.

Contribution

It introduces a Hamiltonian structure for projective polygon evolutions, linking discrete geometric flows to integrable systems and extending these concepts to higher dimensions.

Findings

Explicit formulas for invariant evolutions of projective polygons.

Establishment of a Hamiltonian structure on invariants.

Proof of complete integrability in planar and higher dimensions.

Abstract

In this paper we describe a well-chosen discrete moving frame and their associated invariants along projective polygons in $\RP^n$, and we use them to write explicit general expressions for invariant evolutions of projective $N$-gons. We then use a reduction process inspired by a discrete Drinfeld-Sokolov reduction to obtain a natural Hamiltonian structure on the space of projective invariants, and we establish a close relationship between the projective $N$-gon evolutions and the Hamiltonian evolutions on the invariants of the flow. We prove that {any} Hamiltonian evolution is induced on invariants by an evolution of $N$-gons - what we call a projective realization - and we give the direct connection. Finally, in the planar case we provide completely integrable evolutions (the Boussinesq lattice related to the lattice $W_3$-algebra), their projective realizations and their Hamiltonian pencil. We generalize both structures to $n$-dimensions and we prove that they are Poisson. We define explicitly the $n$-dimensional generalization of the planar evolution (the discretization of the $W_n$-algebra) and prove that it is completely integrable, providing also its projective realization.