Infinite-dimensional Lagrange-Dirac systems with boundary energy flow I: Foundations
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914143024447488 |
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| author | Gay-Balmaz, François Abella, Álvaro Rodríguez Yoshimura, Hiroaki |
| author_facet | Gay-Balmaz, François Abella, Álvaro Rodríguez Yoshimura, Hiroaki |
| contents | A new geometric approach to systems with boundary energy flow is developed using infinite-dimensional Dirac structures within the Lagrangian formalism. This framework satisfies a list of consistency criteria with the geometric setting of finite-dimensional mechanics. In particular, the infinite-dimensional Dirac structure can be constructed from the canonical symplectic form on the system's phase space; the system's evolution equations can be derived equivalently from either a variational perspective or a Dirac structure perspective; the variational principle employed is a direct extension of Hamilton's principle in classical mechanics; and the approach allows for a process of system interconnection within its formulation. This is achieved by developing an appropriate infinite dimensional version of the previously developed Lagrange-Dirac dynamical systems. A key step in this construction is the careful choice of a suitable dual space to the configuration space, specifically, a subspace of the topological dual that captures the system's behavior in both the interior and the boundary, while allowing for a natural extension of the canonical geometric structures of mechanics. This paper focuses on systems where the configuration space consists of differential forms on a smooth manifold with a boundary. To illustrate our theory, several examples, including nonlinear wave equations, the telegraph equation, and the Maxwell equations are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17551 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinite-dimensional Lagrange-Dirac systems with boundary energy flow I: Foundations Gay-Balmaz, François Abella, Álvaro Rodríguez Yoshimura, Hiroaki Symplectic Geometry Mathematical Physics Dynamical Systems A new geometric approach to systems with boundary energy flow is developed using infinite-dimensional Dirac structures within the Lagrangian formalism. This framework satisfies a list of consistency criteria with the geometric setting of finite-dimensional mechanics. In particular, the infinite-dimensional Dirac structure can be constructed from the canonical symplectic form on the system's phase space; the system's evolution equations can be derived equivalently from either a variational perspective or a Dirac structure perspective; the variational principle employed is a direct extension of Hamilton's principle in classical mechanics; and the approach allows for a process of system interconnection within its formulation. This is achieved by developing an appropriate infinite dimensional version of the previously developed Lagrange-Dirac dynamical systems. A key step in this construction is the careful choice of a suitable dual space to the configuration space, specifically, a subspace of the topological dual that captures the system's behavior in both the interior and the boundary, while allowing for a natural extension of the canonical geometric structures of mechanics. This paper focuses on systems where the configuration space consists of differential forms on a smooth manifold with a boundary. To illustrate our theory, several examples, including nonlinear wave equations, the telegraph equation, and the Maxwell equations are presented. |
| title | Infinite-dimensional Lagrange-Dirac systems with boundary energy flow I: Foundations |
| topic | Symplectic Geometry Mathematical Physics Dynamical Systems |
| url | https://arxiv.org/abs/2501.17551 |