Contact Dynamics: Legendrian and Lagrangian Submanifolds

Oğul Esen, Manuel Lainz Valcázar, Manuel de León and Juan Carlos Marrero
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Oğul Esen: Department of Mathematics, Gebze Technical University, Gebze 41400, Turkey
Manuel Lainz Valcázar: Campus Cantoblanco Consejo Superior de Investigaciones Científicas C/Nicolás Cabrera, Instituto de Ciencias Matematicas, 13–15, 28049 Madrid, Spain
Manuel de León: Campus Cantoblanco Consejo Superior de Investigaciones Científicas C/Nicolás Cabrera, Instituto de Ciencias Matematicas, 13–15, 28049 Madrid, Spain
Juan Carlos Marrero: ULL-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, Spain

Mathematics, 2021, vol. 9, issue 21, 1-41

Abstract: 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.

Keywords: Tulczyjew’s triple; contact dynamics; evolution contact dynamics; Legendrian submanifold; Lagrangian submanifold (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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