Local bi-integrability of bi-Hamiltonian systems, Part II: Real smooth case
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910674547900416 |
|---|---|
| author | Kozlov, I. K. |
| author_facet | Kozlov, I. K. |
| contents | We prove that any bi-Hamiltonian system $v = \left(\mathcal{A} + λ\mathcal{B}\right)dH_λ$ on a real smooth manifold that is Hamiltonian with respect all Poisson brackets $\left(\mathcal{A} + λ\mathcal{B}\right)$ is locally bi-integrable. We construct a complete set of functions $\mathcal{G}$ in bi-involution by extending the set of standard integrals $\mathcal{F}$ consisting of Casimir functions of Poisson brackets, eigenvalues of the Poisson pencil, and the Hamiltonians. Moreover, we show that at a generic point of $M$ differentials of the extended family $d \mathcal{G}$ can realize any bi-Lagrangian subspace $L$ containing the differentials of the standard integrals $d \mathcal{F}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21642 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local bi-integrability of bi-Hamiltonian systems, Part II: Real smooth case Kozlov, I. K. Symplectic Geometry Differential Geometry We prove that any bi-Hamiltonian system $v = \left(\mathcal{A} + λ\mathcal{B}\right)dH_λ$ on a real smooth manifold that is Hamiltonian with respect all Poisson brackets $\left(\mathcal{A} + λ\mathcal{B}\right)$ is locally bi-integrable. We construct a complete set of functions $\mathcal{G}$ in bi-involution by extending the set of standard integrals $\mathcal{F}$ consisting of Casimir functions of Poisson brackets, eigenvalues of the Poisson pencil, and the Hamiltonians. Moreover, we show that at a generic point of $M$ differentials of the extended family $d \mathcal{G}$ can realize any bi-Lagrangian subspace $L$ containing the differentials of the standard integrals $d \mathcal{F}$. |
| title | Local bi-integrability of bi-Hamiltonian systems, Part II: Real smooth case |
| topic | Symplectic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2410.21642 |