Dissipation in Lagrangian Formalism

Abstract

In this paper, we present a method by which it is possible to describe a dissipative system (that is modeled by a linear differential equation) in Lagrangian formalism, without the trouble of finding the proper way to model the environment. The concept of the presented method is to create a function that generates the measurable physical quantity, similarly to electrodynamics, where the scalar potential and vector potential generate the electric and magnetic fields. The method is examined in the classical case; the question of quantization is unanswered.

Dynamic Equilibrium Equations in Unified Mechanics Theory

“…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. It is stated in the literature [38] that the numerical solutions to the differential equations formulated by the presented [38] method would yield non-physical solutions, if not stabilized by appropriate conditions.…”

“…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. It is stated in the literature [38] that the numerical solutions to the differential equations formulated by the presented [38] method would yield non-physical solutions, if not stabilized by appropriate conditions. However, the method [38] did not give any physical basis in incorporating a potential in the Lagrangian.…”

“…It is stated in the literature [38] that the numerical solutions to the differential equations formulated by the presented [38] method would yield non-physical solutions, if not stabilized by appropriate conditions. However, the method [38] did not give any physical basis in incorporating a potential in the Lagrangian. None of the above approaches brought the idea to describe dissipation within Lagrangian formalism based on the second law of thermodynamics.…”

“…We define the factor, ψ as the ratio between the maximum entropy at which TSI, Φ tends to unity (say, at Φ = 0.99) and the maximum entropy that the mass-spring system could generate, for an initial excitation, if it had zero in-efficiency. Hence, by substituting Equation (38) in Equation 34, we get the following equation,…”

“…The proposed Equation (39), is solved together with Equations (38) and (43) and the results are presented in the following section.…”