Integrable evolutions of twisted polygons in centro-affine $\mathbb{R}^m$
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2019
- DOI:
- 10.48550/arXiv.1909.13435
- arXiv:
- arXiv:1909.13435
- Bibcode:
- 2019arXiv190913435M
- Keywords:
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- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- Mathematics - Differential Geometry;
- 37K10;
- 37K25;
- 53C44
- E-Print:
- 39 pages