Contact Dynamics: Legendrian and Lagrangian Submanifolds
We are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-10-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/9/21/2704 |
| _version_ | 1797512114738298880 |
|---|---|
| author | Oğul Esen Manuel Lainz Valcázar Manuel de León Juan Carlos Marrero |
| author_facet | Oğul Esen Manuel Lainz Valcázar Manuel de León Juan Carlos Marrero |
| author_sort | Oğul Esen |
| collection | DOAJ |
| description | We are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two different generators of the same Legendrian submanifold. A variant of contact Tulczyjew’s triple is constructed for evolution contact dynamics. |
| first_indexed | 2024-03-10T05:57:19Z |
| format | Article |
| id | doaj.art-eda3043d3612424c80d47ab12514132e |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T05:57:19Z |
| publishDate | 2021-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-eda3043d3612424c80d47ab12514132e2023-11-22T21:17:37ZengMDPI AGMathematics2227-73902021-10-01921270410.3390/math9212704Contact Dynamics: Legendrian and Lagrangian SubmanifoldsOğul Esen0Manuel Lainz Valcázar1Manuel de León2Juan Carlos Marrero3Department of Mathematics, Gebze Technical University, Gebze 41400, TurkeyCampus Cantoblanco Consejo Superior de Investigaciones Científicas C/Nicolás Cabrera, Instituto de Ciencias Matematicas, 13–15, 28049 Madrid, SpainCampus Cantoblanco Consejo Superior de Investigaciones Científicas C/Nicolás Cabrera, Instituto de Ciencias Matematicas, 13–15, 28049 Madrid, SpainULL-CSIC Geometria Diferencial y Mecánica Geométrica, Departamento de Matematicas, Estadistica e I O, Sección de Matemáticas, Facultad de Ciencias, Universidad de la Laguna, 38071 La Laguna, SpainWe are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two different generators of the same Legendrian submanifold. A variant of contact Tulczyjew’s triple is constructed for evolution contact dynamics.https://www.mdpi.com/2227-7390/9/21/2704Tulczyjew’s triplecontact dynamicsevolution contact dynamicsLegendrian submanifoldLagrangian submanifold |
| spellingShingle | Oğul Esen Manuel Lainz Valcázar Manuel de León Juan Carlos Marrero Contact Dynamics: Legendrian and Lagrangian Submanifolds Mathematics Tulczyjew’s triple contact dynamics evolution contact dynamics Legendrian submanifold Lagrangian submanifold |
| title | Contact Dynamics: Legendrian and Lagrangian Submanifolds |
| title_full | Contact Dynamics: Legendrian and Lagrangian Submanifolds |
| title_fullStr | Contact Dynamics: Legendrian and Lagrangian Submanifolds |
| title_full_unstemmed | Contact Dynamics: Legendrian and Lagrangian Submanifolds |
| title_short | Contact Dynamics: Legendrian and Lagrangian Submanifolds |
| title_sort | contact dynamics legendrian and lagrangian submanifolds |
| topic | Tulczyjew’s triple contact dynamics evolution contact dynamics Legendrian submanifold Lagrangian submanifold |
| url | https://www.mdpi.com/2227-7390/9/21/2704 |
| work_keys_str_mv | AT ogulesen contactdynamicslegendrianandlagrangiansubmanifolds AT manuellainzvalcazar contactdynamicslegendrianandlagrangiansubmanifolds AT manueldeleon contactdynamicslegendrianandlagrangiansubmanifolds AT juancarlosmarrero contactdynamicslegendrianandlagrangiansubmanifolds |