On the Lagrangian description of dissipative systems

Abstract

We consider the Lagrangian formulation with duplicated variables of dissipative mechanical systems. The application of Noether theorem leads to physical observable quantities which are not conserved, like energy and angular momentum, and conserved quantities like the Hamiltonian, that generate symmetry transformations and do not correspond to observables. We show that there are simple relations among the equations satisfied by these two types of quantities. In the case of the damped harmonic oscillator, from the quantities obtained by Noether theorem follows the algebra of Feshbach and Tikochinsky. Further, if we consider the whole dynamics, the degrees of freedom separate into a physical and an unphysical sector. We analyze several cases, with linear and nonlinear dissipative forces; the physical consistency of the solutions is ensured observing that the unphysical sector has always the trivial solution.PACS numbers: 04.20.Fy,12.60.Jv * Electronic address: nephtalieliceo@hotmail.com † Electronic address: cramirez@fcfm.buap.mx 2 Hamilton variational principle. In this approach the variation of the action is done with boundary conditions only at the initial time, independently for each of both variables, and at the final time these variables must coincide. A similar development for classical and quantum mechanics was given by means of an extension of the Closed Time Path formalism to classical mechanics by Polonyi [40,41], who in [42] considers the issue of breaking of time reversal symmetry.The main interest in the study of phenomenological dissipative systems is on their quantum description. Classically, the doubled variable formalism allows to write the equations of motion and after that the additional variables are somehow discarded. In fact, these variables are considered as an artifice which takes account of the dissipative external influence, the whole system being isolated. However, from its construction, the nonconservative Lagrangian has not the standard form due to the time reversed characteristics of the additional sector, i.e. the kinetic term is not positive definite and the potential appears with an unstable term. Thus, an interpretation of its outcome as a whole is not obvious. On the other side, in a quantum theory every interacting degree of freedom in general contributes to the probabilities, spectra and mean values, as they form part of the operator algebra. Thus, it would be desirable to consider the classical theory taking into account the whole dynamics. Moreover, a general knowledge of the relevant quantities in the theory, as delivered e.g. by Noether theorem, is necessary for the definition of the Hilbert space. Actually, in the doubled variable approach, Noether theorem has been applied considering the conservation laws of the conservative part, and these laws are violated due to the dissipative terms [41,43]. Furthermore, Noether theorem has been applied in similar approaches to the symmetries of the whole doubled variables action in [44,45], and for time dependent lagrangians in [46]. ...

Hyperbolic ring based formulation for thermo field dynamics, quantum dissipation, entanglement, and holography

“…Thus, the hyperbolic scheme provides an appropriate and alternative description of the classical and quantum dissipative dynamics, and in principle a broadly applicable theoretical technique, for example for studying and reformulating the different dissipative systems described previously in both high energy and condensed matter scenarios. The main virtues of the scheme developed here are, for example, to cure the pathologies associated with existence of many unitarity inequivalent representations, and to cure the nonpreservation of the field commutators due to the presence of damping factors, which have been persistent problems in the traditional treatments on dissipative dynamics (see [20], and references cited therein); additionally within the present scheme the field commutators depend in a novel way on the dissipative parameter and on the background dimension, since they do not correspond simply to Dirac delta distributions; similar results are obtained for physical observables and operators. As an explicit example, the case of 1 + 1 dissipative quantum field theory, of wide interest in condensed matter systems of low dimensionality, will be developed in exact form, and it will show profound differences with respect to the conventional treatments.…”

“…Due to this fact, and although the formulations of the dissipative systems have been widely studied, all treatments have suffered historically from this problem; in fact, in the traditional schemes there exists a physical mechanism for preserving the canonical structure, the fluctuating forces [36]. For an alternative description of dissipative quantum mechanics, and an updating on the different formulations to the date, describing those long-lived problems, see for example [20].…”

Dynamic Equilibrium Equations in Unified Mechanics Theory

“…Even though the approach [36] is consistent with the variational principles, the Riemann-Liouville fractional derivatives in the Lagrangian, were introduced without any thermodynamic basis. Mart´ınez-P´erez and Ram´ırez [37] studied the doubled variable Lagrangian formulation of dissipative mechanical systems by the application of the Noether theorem. Szegleti and Márkus [38] presented a method which creates a potential function that generates the measurable physical quantity, to describe a dissipative system modeled by a linear differential equation in Lagrangian formalism.…”

Macroscopic Limit of Quantum Systems

“…The formalism to follow the macroscopical limit of quantum systems should be a CQCO scheme: it should handle classical (C) [51][52][53][54][55] and quantum (Q) [56][57][58][59] dynamics because the macroscopical limit implies classical physics should further cover closed (C) and open (O) dynamics since a macroscopic system cannot be kept isolated on the microscopic scale. These possibilities are offered by the CTP formalism, initially proposed to deal with the perturbation expansion in the Heisenberg representation [60] and with non-equilibrium phenomena in many-body systems [61].…”