New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries

Abstract

We provide new insights into the contact Hamiltonian and Lagrangian formulations of dissipative mechanical systems. In particular, we state a new form of the contact dynamical equations, and we review two recently presented Lagrangian formalisms, studying their equivalence. We define several kinds of symmetries for contact dynamical systems, as well as the notion of dissipation laws, prove a dissipation theorem and give a way to construct conserved quantities. Some well-known examples of dissipative systems are discussed.

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,…”

“…Contact geometry was introduced in Sophus Lie's study of differential equations, and has been the subject of intense research, especially related to low-dimensional topology [8]. In recent years, contact Hamiltonian systems have found many applications, first in the context of thermodynamics [9][10][11] and, more recently, in the context of the Hamiltonisation of several dissipative dynamical systems [12][13][14][15][16][17][18][19]. The large number of applications of contact systems that have appeared recently motivated research on geometric numerical integration [15,16,20,21].…”

From Geometry to Coherent Dissipative Dynamics in Quantum Mechanics

“…This is based on the symplectic formulation of quantum mechanics and due to the analogy with the geometric description of classical dissipative systems: In classical mechanics, one can describe a wide class of dissipative systems by referring to the contactification of the symplectic phase space and then using contact Hamiltonian systems to define the dynamics. It has been shown that this approach, when applicable, provides several positive features, such as relying on canonical variables and producing a generalization of canonical transformations [7], enabling an extension of both Liouville and Noether's theorems to the dissipative case [8][9][10][11][12][13], and providing a description in terms of variational principles [14][15][16][17][18][19][20] together with a natural route to field theories with dissipation [21].…”

“…In this section, we consider a particular class of dissipative quantum systems, which are those that admit a contact Hamiltonian description (see also [7,11,12,21,26,[36][37][38][39] for detailed discussions on the strengths and limitations of this approach both in the classical and quantum settings).…”

Contact Hamiltonian and Lagrangian systems with nonholonomic constraints

“…Nevertheless, there is a natural geometric description for these systems based on contact geometry. Contact Hamiltonian systems have attracted a lot of attention in the last years; the dynamics of singular Lagrangian systems [15], symmetries and dissipated quantities [14,20], applications to thermodynamics [42,44], discrete dynamics [41,45], among others.…”