Contact Hamiltonian mechanics

Abstract

In this work we introduce contact Hamiltonian mechanics, an extension of symplectic Hamiltonian mechanics, and show that it is a natural candidate for a geometric description of non-dissipative and dissipative systems. For this purpose we review in detail the major features of standard symplectic Hamiltonian dynamics and show that all of them can be generalized to the contact case. (Alessandro Bravetti), hans@ciencias.unam.mx (Hans Cruz), diego.tapias@nucleares.unam.mx (Diego Tapias) 4 Conclusions and perspectives 29 Appendix A Invariants for the damped parametric oscillator 32 Appendix B Equivalence between the contact Hamilton-Jacobi equation and the contact Hamiltonian equations 34

Energy consumption for shortcuts to adiabaticity

“…It was soon clear that this quantity could not represent all relevant energy flows, which led to the consideration of alternative measures [7]. Several disparate definitions of energy cost have been proposed in the context of quantum thermodynamics to characterize quantum engines and refrigerators [8][9][10][11][12][13][14][15][16][17][18]. These definitions have been systematically formulated in terms of the cycling system (PS) alone.…”

Contact Hamiltonian Dynamics: The Concept and Its Use

“…In this sense, a naive proposal has been sketched already in [10], whereas in [48][49][50], one can find more formal approaches.…”

“…In particular, when H does not depend on z, Equations (7) and (8) give exactly Hamilton's equations in the symplectic sub-space parametrized by x a and y a . Finally, (9) in this case is the usual definition of Hamilton's principal function [10,14]. An important difference between contact Hamiltonian dynamics and symplectic Hamiltonian dynamics is that in the contact case, the Hamiltonian H is not preserved along the evolution.…”

“…In [10], it has been shown that considering the mechanical phase space of a system to be a contact manifold (T CM , η CM ), the contact Hamiltonian Equations (7)-(9) extend the standard Hamiltonian equations to include cases where dissipation is considered. As an example, considering the (on-dimensional) contact Hamiltonian:…”

“…For instance, contact transformations generalize canonical transformations, and one can find a contact version of the Hamilton-Jacobi theory, equivalent to the dynamics (7)-(9) (cf. [10,46]). …”