Foundations on k-contact geometry
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866916598231597056 |
|---|---|
| author | de Lucas, Javier Rivas, Xavier Sobczak, Tomasz |
| author_facet | de Lucas, Javier Rivas, Xavier Sobczak, Tomasz |
| contents | k-Contact geometry is a generalisation of contact geometry to analyse field theories. We develop an approach to k-contact geometry based on distributions that are distributionally maximally non-integrable and admit, locally, k commuting supplementary Lie symmetries: the k-contact distributions. We related k-contact distributions with Engel, Goursat and other distributions, which have mathematical and physical interest. We give necessary topological conditions for the existence of globally defined Lie symmetries, k-contact Lie groups are defined and studied, and we study and propose a k-contact Weinstein conjecture for co-oriented k-contact manifolds. Polarisations for k-contact distributions are introduced and it is shown that a polarised k-contact distribution is locally diffeomorphic to the Cartan distribution of the first-order jet bundle over a fibre bundle of order k. We relate k-contact manifolds to presymplectic and k-symplectic manifolds on fibre bundles of larger dimension and define types of submanifolds in k-contact geometry. We study Hamilton-De Donder-Weyl equations in Lie groups for the first time. A theory of k-contact Hamiltonian vector fields is developed, and we describe characteristics of Lie symmetries for first-order partial differential equations in a k-contact Hamiltonian manner. We use our techniques to analyse Hamilton-Jacobi and Dirac equations. Other potential applications of k-contact distributions to non-holonomic and control systems are briefly described. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11001 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Foundations on k-contact geometry de Lucas, Javier Rivas, Xavier Sobczak, Tomasz Differential Geometry Mathematical Physics Analysis of PDEs 53C15, 53C12, 58A30, 53C10, 53D10, 58J60 k-Contact geometry is a generalisation of contact geometry to analyse field theories. We develop an approach to k-contact geometry based on distributions that are distributionally maximally non-integrable and admit, locally, k commuting supplementary Lie symmetries: the k-contact distributions. We related k-contact distributions with Engel, Goursat and other distributions, which have mathematical and physical interest. We give necessary topological conditions for the existence of globally defined Lie symmetries, k-contact Lie groups are defined and studied, and we study and propose a k-contact Weinstein conjecture for co-oriented k-contact manifolds. Polarisations for k-contact distributions are introduced and it is shown that a polarised k-contact distribution is locally diffeomorphic to the Cartan distribution of the first-order jet bundle over a fibre bundle of order k. We relate k-contact manifolds to presymplectic and k-symplectic manifolds on fibre bundles of larger dimension and define types of submanifolds in k-contact geometry. We study Hamilton-De Donder-Weyl equations in Lie groups for the first time. A theory of k-contact Hamiltonian vector fields is developed, and we describe characteristics of Lie symmetries for first-order partial differential equations in a k-contact Hamiltonian manner. We use our techniques to analyse Hamilton-Jacobi and Dirac equations. Other potential applications of k-contact distributions to non-holonomic and control systems are briefly described. |
| title | Foundations on k-contact geometry |
| topic | Differential Geometry Mathematical Physics Analysis of PDEs 53C15, 53C12, 58A30, 53C10, 53D10, 58J60 |
| url | https://arxiv.org/abs/2409.11001 |