On the geometry of Lagrangian one-forms
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913765030625280 |
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| author | Caudrelier, Vincent Harland, Derek |
| author_facet | Caudrelier, Vincent Harland, Derek |
| contents | Lagrangian multiform theory is a variational framework for integrable systems. In this article we introduce a new formulation which is based on symplectic geometry and which treats position, momentum and time coordinates of a finite-dimensional integrable hierarchy on an equal footing. This formulation allows a streamlined one-step derivation of both the multi-time Euler-Lagrange equations and the closure relation (encoding integrability). We argue that any Lagrangian one-form for a finite-dimensional system can be recast in our new framework. This framework easily extends to non-commuting flows and we show that the equations characterising (infinitesimal) Hamiltonian Lie group actions are variational in character. We reinterpret these equations as a system of compatible non autonomous Hamiltonian equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14700 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the geometry of Lagrangian one-forms Caudrelier, Vincent Harland, Derek Mathematical Physics High Energy Physics - Theory Symplectic Geometry Exactly Solvable and Integrable Systems Lagrangian multiform theory is a variational framework for integrable systems. In this article we introduce a new formulation which is based on symplectic geometry and which treats position, momentum and time coordinates of a finite-dimensional integrable hierarchy on an equal footing. This formulation allows a streamlined one-step derivation of both the multi-time Euler-Lagrange equations and the closure relation (encoding integrability). We argue that any Lagrangian one-form for a finite-dimensional system can be recast in our new framework. This framework easily extends to non-commuting flows and we show that the equations characterising (infinitesimal) Hamiltonian Lie group actions are variational in character. We reinterpret these equations as a system of compatible non autonomous Hamiltonian equations. |
| title | On the geometry of Lagrangian one-forms |
| topic | Mathematical Physics High Energy Physics - Theory Symplectic Geometry Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2412.14700 |