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Contact Dynamics: Legendrian and Lagrangian Submanifolds

Author

Listed:
  • 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
    Real Academia de Ciencias. C/Valverde, 22, 28004 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)

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.

Suggested Citation

  • Oğul Esen & Manuel Lainz Valcázar & Manuel de León & Juan Carlos Marrero, 2021. "Contact Dynamics: Legendrian and Lagrangian Submanifolds," Mathematics, MDPI, vol. 9(21), pages 1-41, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:21:p:2704-:d:664075
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