  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.
