On the lagrangian description of dissipative systems

TL;DR

This paper explores a Lagrangian approach with duplicated variables to describe dissipative systems, revealing relations between conserved and non-conserved quantities, and analyzing the physical and unphysical sectors in various cases.

Contribution

It introduces a Lagrangian formulation for dissipative systems with duplicated variables, clarifies the role of conserved and non-conserved quantities, and analyzes the physical consistency of solutions.

Findings

Noether theorem yields observable and non-observable quantities.

The algebra of Feshbach and Tikochinsky emerges from this framework.

Unphysical degrees of freedom are always trivial, ensuring physical consistency.

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.