Contact Hamiltonian systems

Abstract

In this paper, we study Hamiltonian systems on contact manifolds, which is an appropriate scenario to discuss dissipative systems. We show how the dissipative dynamics can be interpreted as a Legendrian submanifold, and also prove a coisotropic reduction theorem similar to the one in symplectic mechanics; as a consequence, we get a method to reduce the dynamics of contact Hamiltonian systems.

Contact Hamiltonian and Lagrangian systems with nonholonomic constraints

“…In the case of contact manifolds, the map Λ can be written directly in terms of the contact structure [13,Section 3] as:…”

“…Contrary to the symplectic case, the energy and the phase space volume are not conserved (see [13,7]).…”

“…Some authors define a contact structure as a smooth distribution D on M such that it is locally generated by contact forms η via the kernels ker η. This definition is not equivalent to ours, and is less convenient for our purposes, as explained in [13].…”

The Hamilton–Jacobi Theory for Contact Hamiltonian Systems

“…Consider a contact manifold [12][13][14][15][16][17] (M, η) with contact form η; this means that η ∧ dη n = 0, and M has odd dimension 2n + 1. Then, there exists a unique vector field R (called Reeb vector field) such that…”

Geometric Numerical Integration of Liénard Systems via a Contact Hamiltonian Approach

“…We refer the reader to [11,13,17,24,25] for further details. For our scope, it will be important in the following to have an explicit expression for the Jacobi bracket of monomial functions, that is,…”