A review on contact Hamiltonian and Lagrangian systems

Abstract

Contact Hamiltonian dynamics is a subject that has still a short history, but with relevant applications in many areas: thermodynamics, cosmology, control theory, and neurogeometry, among others. In recent years there has been a great effort to study this type of dynamics both in theoretical aspects and in its potential applications in geometric mechanics and mathematical physics. This paper is intended to be a review of some of the results that the authors and their collaborators have recently obtained on the subject.

Hamiltonian systems on almost cosymplectic manifolds

“…The gradient vector field grad H, the Hamiltonian vector field X H and the evolution vector field E H attached to the function H are defined as solutions of the equations: [41,42]: contact Hamiltonian system -is defined by (M, η), where M is a (2n + 1)-dimensional manifold, η ∈ D If in equations (5.16) and (5.17) we neglect the "green parts", we get X H (5.3) on the symplectic manifold (M, Ω).…”

“…Such a manifold could be called generalized transitive almost cosymplectic space. We also endow XJ 1 with a contact structure in the sens of [41].…”

On Geometrical Couplings of Dissipation and Curl Forces

“…The Herglotz principle yields a variational description of nonconservative as well as conservative processes involving one independent variable. The Herglotz principle is defined by the functional z(q; τ ) through a differential equation of the form [20,27,39] ż = L(t, q i , qi , z),…”

From geometry to coherent dissipative dynamics in quantum mechanics

“…In this section we consider a particular class of dissipative quantum systems, those that admit a contact Hamiltonian description (see also [7,12,19,22,23,26,27,34,51] for detailed discussions on the strengths and limitations of this approach both in the classical and quantum settings).…”