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Integrable evolutions of twisted polygons in centro-affine $\mathbb{R}^m$

Gloria Marí Beffa, Annalisa Calini

Abstract

We show that discrete $W_m$ lattices are bi-Hamiltonian, using geometric realizations of discretizations of the Adler-Gel'fand-Dikii flows as local evolutions of arc length-parametrized polygons in centro-affine space. We prove the compatibility of two known Hamiltonian structure defined on the space of geometric invariants by lifting them to a pair of pre-symplectic forms on the space of arc length parametrized polygons. The simplicity of the expressions of the pre-symplectic forms makes the proof of compatibility straightforward. We also study their kernels and possible integrable systems associated to the pair.

Integrable evolutions of twisted polygons in centro-affine $\mathbb{R}^m$

Abstract

We show that discrete lattices are bi-Hamiltonian, using geometric realizations of discretizations of the Adler-Gel'fand-Dikii flows as local evolutions of arc length-parametrized polygons in centro-affine space. We prove the compatibility of two known Hamiltonian structure defined on the space of geometric invariants by lifting them to a pair of pre-symplectic forms on the space of arc length parametrized polygons. The simplicity of the expressions of the pre-symplectic forms makes the proof of compatibility straightforward. We also study their kernels and possible integrable systems associated to the pair.

Paper Structure

This paper contains 16 sections, 29 theorems, 249 equations.

Table of Contents

  1. Introduction
  2. Background: Evolutions of Polygons in Centro-affine Space
  3. Evolutions of Polygons and Invariants
  4. Hamiltonian structures on the moduli space of polygons
  5. A twisted Poisson structure on a Poisson-Lie group
  6. Poisson structures on the moduli space of centro-affine polygons
  7. Polygon evolutions and Hamiltonian flows in the moduli space
  8. Evolution of moving frames.
  9. Poisson brackets
  10. Invariant Hamiltonian flows
  11. Lifting Hamiltonian structures to the moduli space of twisted polygons
  12. Example ($m=3$)
  13. The general case.
  14. The kernels of the pre-symplectic forms
  15. The kernel of $\omega_1$
  16. ...and 1 more sections

Key Result

Proposition 3.3

If $N$ and $m$ are co-prime, the moduli space of $\mathcal{M}_N^1$ is diffeomorphic to the moduli space of non-degenerate twisted projective polygons in $\mathbb{RP}^{m-1}$ with the same monodromy.

Theorems & Definitions (71)

  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4: Left and right gradients
  • Remark
  • Theorem 3.5
  • Remark
  • Proposition 3.6
  • proof
  • ...and 61 more