Variational Dissipative Mechanics on Lie Algebroids
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911327972229120 |
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| author | Simoes, Alexandre Anahory Colombo, Leonardo |
| author_facet | Simoes, Alexandre Anahory Colombo, Leonardo |
| contents | We formulate a Herglotz-type variational principle on a Lie algebroid and derive the corresponding Euler--Lagrange--Herglotz equations for a Lagrangian depending on an additional scalar variable $z$. This provides a geometric framework for dissipative systems on Lie algebroids and recovers, as special cases, the classical Euler--Lagrange--Herglotz equations on tangent bundles, the Euler--Poincaré--Herglotz equations on a Lie algebra, and the Lagrange--Poincaré--Herglotz equations on Atiyah algebroids of principal bundles. Starting from the local formulation, we then use Lie algebroid connections to obtain a coordinate-free Euler--Lagrange--Poincaré--Herglotz and Hamilton--Pontryagin--Herglotz theory. Finally, we establish energy balance laws and Noether--Herglotz-type results, in which classical conserved quantities are replaced by dissipated invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17424 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Variational Dissipative Mechanics on Lie Algebroids Simoes, Alexandre Anahory Colombo, Leonardo Mathematical Physics Differential Geometry Dynamical Systems We formulate a Herglotz-type variational principle on a Lie algebroid and derive the corresponding Euler--Lagrange--Herglotz equations for a Lagrangian depending on an additional scalar variable $z$. This provides a geometric framework for dissipative systems on Lie algebroids and recovers, as special cases, the classical Euler--Lagrange--Herglotz equations on tangent bundles, the Euler--Poincaré--Herglotz equations on a Lie algebra, and the Lagrange--Poincaré--Herglotz equations on Atiyah algebroids of principal bundles. Starting from the local formulation, we then use Lie algebroid connections to obtain a coordinate-free Euler--Lagrange--Poincaré--Herglotz and Hamilton--Pontryagin--Herglotz theory. Finally, we establish energy balance laws and Noether--Herglotz-type results, in which classical conserved quantities are replaced by dissipated invariants. |
| title | Variational Dissipative Mechanics on Lie Algebroids |
| topic | Mathematical Physics Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2512.17424 |