Contact Dynamics versus Legendrian and Lagrangian Submanifolds

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.

Implicit Contact Dynamics and Hamilton-Jacobi Theory

“…The kernel of the contact one-form η Q determines a symplectic subbundle HT * Q of the tangent bundle T T * Q. Moreover, the product space HT * Q×R turns out to be a symplectic manifold, [20]. This is stated in Theorem 4.1 in Subsection 4.1.…”

“…Morse families are generating Legendrian submanifolds of the extended cotangent manifolds T * Q. To transfer such a Legendrian submanifold to a contact manifold (M, η) one needs to employ a special contact structure, see [20]. Let us first introduce the notion of the special contact structure and then merge it with a Morse family.…”

“…Consider the contact manifold (T * Q, η Q ). We cite [20] for all of the results in this Subsection. In (2.2), we have stated that the kernel of the contact one-form is a symplectic vector bundle of the tangent bundle.…”

“…Referring to the Morse family E = E(q, p, z, q) in (5.3) defined on the Whitney sum T * Q× Q×R T Q and to the generic formulation in (4.30) one arrives at the Lagrangian submanifold that determines the evolution Herglotz equations (5.9), see [20].…”

“…defined on the Whitney sum T * Q × Q×R T Q over the base manifold T * Q. Here, the velocity qi is the Lagrange multiplier in the Morse family E. By substituting E into the generic formulation in (3.25) one arrives at the Legendrian submanifold considering the Herglotz equations (5.1), see[20] …”