Local bi-integrability of bi-Hamiltonian systems via bi-Poisson reduction
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929562485522432 |
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| author | Kozlov, I. K. |
| author_facet | Kozlov, I. K. |
| contents | We prove that any bi-Hamiltonian system $v = \left(\mathcal{A} + λ\mathcal{B}\right)dH_λ$ that is Hamiltonian with respect all Poisson brackets $\mathcal{A} + λ\mathcal{B}$ is locally bi-integrable in both the real smooth case, when all eigenvalues of the Poisson pencil $\mathcal{P} = \left\{\mathcal{A} + λ\mathcal{B}\right\}$ are real, and in the complex analytic case. A complete set of functions in bi-involution is constructed by extending the set of standard integrals, which consists of Casimir functions of Poisson brackets, eigenvalues of the Poisson pencil and Hamiltonians. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20574 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local bi-integrability of bi-Hamiltonian systems via bi-Poisson reduction Kozlov, I. K. Symplectic Geometry Differential Geometry We prove that any bi-Hamiltonian system $v = \left(\mathcal{A} + λ\mathcal{B}\right)dH_λ$ that is Hamiltonian with respect all Poisson brackets $\mathcal{A} + λ\mathcal{B}$ is locally bi-integrable in both the real smooth case, when all eigenvalues of the Poisson pencil $\mathcal{P} = \left\{\mathcal{A} + λ\mathcal{B}\right\}$ are real, and in the complex analytic case. A complete set of functions in bi-involution is constructed by extending the set of standard integrals, which consists of Casimir functions of Poisson brackets, eigenvalues of the Poisson pencil and Hamiltonians. |
| title | Local bi-integrability of bi-Hamiltonian systems via bi-Poisson reduction |
| topic | Symplectic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2410.20574 |