Integrable evolutions of twisted polygons in centro-affine $\mathbb{R}^m$
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866916449734361088 |
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| author | Beffa, Gloria Marí Calini, Annalisa |
| author_facet | Beffa, Gloria Marí Calini, Annalisa |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_13435 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Integrable evolutions of twisted polygons in centro-affine $\mathbb{R}^m$ Beffa, Gloria Marí Calini, Annalisa Exactly Solvable and Integrable Systems Differential Geometry 37K10, 37K25, 53C44 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. |
| title | Integrable evolutions of twisted polygons in centro-affine $\mathbb{R}^m$ |
| topic | Exactly Solvable and Integrable Systems Differential Geometry 37K10, 37K25, 53C44 |
| url | https://arxiv.org/abs/1909.13435 |