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 (...

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Bibliographic Details
Published in:Mathematics (Basel) 2021-11, Vol.9 (21), p.2704
Main Authors: Esen, Oğul, Lainz Valcázar, Manuel, de León, Manuel, Marrero, Juan Carlos
Format: Article
Language:English
Subjects:
Algorithms
contact dynamics
Dynamics
evolution contact dynamics
Geometry
Isomorphism
Lagrangian submanifold
Legendrian submanifold
Manifolds (mathematics)
Mathematics
Tulczyjew’s triple
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math9212704