Jacobi-Haantjes manifolds, integrability and dissipative mechanical systems
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908726863069184 |
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| author | Azuaje, Rafael Tempesta, Piergiulio |
| author_facet | Azuaje, Rafael Tempesta, Piergiulio |
| contents | The notion of Jacobi-Haantjes manifold, consisting of a Jacobi manifold endowed with an algebra of extended Haantjes operator fields, is proposed as a natural geometric framework which allows us to define the notion of integrability of both conservative and dissipative Hamiltonian systems, in a unified way. As a reduction, contact-Haantjes manifolds are defined. We prove that the integrability of a contact Hamiltonian system is equivalent to the existence of a suitable Abelian extended Haantjes algebra associated with the system. This result allows us to define a large class of new, completely integrable contact Hamiltonian systems from a given extended Haantjes algebra.
Moreover, we propose a theory of separation of variables for dissipative systems. This result is achieved by lifting a dissipative system into a higher-dimensional manifold, obtained as the symplectization of the Jacobi-Haantjes structure associated with the system. This new manifold naturally acquires the structure of a symplectic-Haantjes manifold. We prove that the Darboux-Haantjes coordinates which separate the Hamilton-Jacobi equation of the higher-dimensional symplectic-Haantjes manifold are in fact separation variables for the Hamilton equations associated with the original dissipative system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11715 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Jacobi-Haantjes manifolds, integrability and dissipative mechanical systems Azuaje, Rafael Tempesta, Piergiulio Mathematical Physics Differential Geometry 53A45, 53D10, 70G45, 70H06 The notion of Jacobi-Haantjes manifold, consisting of a Jacobi manifold endowed with an algebra of extended Haantjes operator fields, is proposed as a natural geometric framework which allows us to define the notion of integrability of both conservative and dissipative Hamiltonian systems, in a unified way. As a reduction, contact-Haantjes manifolds are defined. We prove that the integrability of a contact Hamiltonian system is equivalent to the existence of a suitable Abelian extended Haantjes algebra associated with the system. This result allows us to define a large class of new, completely integrable contact Hamiltonian systems from a given extended Haantjes algebra. Moreover, we propose a theory of separation of variables for dissipative systems. This result is achieved by lifting a dissipative system into a higher-dimensional manifold, obtained as the symplectization of the Jacobi-Haantjes structure associated with the system. This new manifold naturally acquires the structure of a symplectic-Haantjes manifold. We prove that the Darboux-Haantjes coordinates which separate the Hamilton-Jacobi equation of the higher-dimensional symplectic-Haantjes manifold are in fact separation variables for the Hamilton equations associated with the original dissipative system. |
| title | Jacobi-Haantjes manifolds, integrability and dissipative mechanical systems |
| topic | Mathematical Physics Differential Geometry 53A45, 53D10, 70G45, 70H06 |
| url | https://arxiv.org/abs/2507.11715 |