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.
Highlights
Dynamics: Legendrian and Lagrangian dynamics are generated by a Lagrangian function defined on the tangent bundle T Q of the configuration space of a physical system, whereas Hamiltonian dynamics are governed by a Hamiltonian function on the cotangent bundle T ∗ Q, which is canonically symplectic [1,2,3,4]
If a Lagrangian function happens to be degenerate, the fiber derivative fails to be a local diffeomorphism since its image space turns out only to be, in the best of cases, a proper submanifold of the cotangent bundle T ∗ Q
By introducing the notion of special contact structure, we have constructed a Tulcyzjew’s triple for contact manifolds, see Diagram (176). This permits us to describe both the contact Lagrangian and the contact Hamiltonian dynamics as Legendrian submanifolds of T T ∗ Q. In this formulation, the Legendre transformation is defined as a passage between two generators of the same Legendrian submanifold
Summary
There is evolution contact Hamiltonian formalism on the extended cotangent bundle In this case, for a Hamiltonian function H, evolution contact Hamilton’s equation is defined to be ι ε H ηQ = 0, Lε H ηQ = dH + R( H )ηQ. The aim of this work is to define Legendre transformation between the Herglotz Equation (7) and the contact Hamilton’s Equation (3) by properly constructing a Tulczyjew’s triple for the case of contact manifolds. By merging special contact geometry with Morse family theory, the Legendre transformation for contact dynamics is defined to be a passage between two different generators of the same Legendrian submanifold. By properly modifying contact Tulczyjew’s triple, the Legendre transformation for the evolution Herglotz equations and the evolution contact Hamilton’s equations are obtained In this theory, contact manifolds and Legendrian submanifolds are replaced by symplectic manifolds and Legendrian submanifolds, respectively.
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