Integrability of bi-Hamiltonian systems using Casimir functions and characteristic polynomials
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912073197289472 |
|---|---|
| author | Kozlov, I. K. |
| author_facet | Kozlov, I. K. |
| contents | In this paper we prove that for a pencil of compatible Poisson brackets $\mathcal{P} = \left\{\mathcal{A} + λ\mathcal{B} \right\}$ the local Casimir functions of Poisson brackets $\mathcal{A} + λ\mathcal{B}$ and coefficients of the characteristic polynomial $p_{\mathcal{P}}$ commute w.r.t. all Poisson brackets of the pencil $\mathcal{P}$. We give a criterion when this family of functions is complete. These results generalize previous constructions of complete commutative subalgebras in the symmetric algebra $S(\mathfrak{g})$ of a finite-dimensional Lie algebra $\mathfrak{g}$ by A.S. Mishchenko & A.T. Fomenko, A.V. Bolsinov & P. Zhang and A.M. Izosimov. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11032 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integrability of bi-Hamiltonian systems using Casimir functions and characteristic polynomials Kozlov, I. K. Symplectic Geometry In this paper we prove that for a pencil of compatible Poisson brackets $\mathcal{P} = \left\{\mathcal{A} + λ\mathcal{B} \right\}$ the local Casimir functions of Poisson brackets $\mathcal{A} + λ\mathcal{B}$ and coefficients of the characteristic polynomial $p_{\mathcal{P}}$ commute w.r.t. all Poisson brackets of the pencil $\mathcal{P}$. We give a criterion when this family of functions is complete. These results generalize previous constructions of complete commutative subalgebras in the symmetric algebra $S(\mathfrak{g})$ of a finite-dimensional Lie algebra $\mathfrak{g}$ by A.S. Mishchenko & A.T. Fomenko, A.V. Bolsinov & P. Zhang and A.M. Izosimov. |
| title | Integrability of bi-Hamiltonian systems using Casimir functions and characteristic polynomials |
| topic | Symplectic Geometry |
| url | https://arxiv.org/abs/2410.11032 |