Is a Dissipative System Always a Gradient System or a Gradient Like System?

TL;DR

The paper develops a differential-forms framework to address whether every dissipative system admits a gradient-like description. By associating a $p=1$ differential form with the dynamics and decomposing it into exact (potential) and residual parts via the $k$-operator, the authors show that closedness of the form implies a potential, while non-closed forms resist closure under coordinate changes. They introduce a generalized nonlinear change of variables, governed by a matrix function $oldsymbol{D}(oldsymbol{y})$ with a nonlinear consistency condition, to transform non-closed cases into closed ones in the new coordinates; this yields a practical route to obtain a potential for several physically relevant systems, notably two-dimensional Josephson junction arrays. Across examples and appendices, they argue that even when an analytic potential is hard to obtain, the potential exists in principle, and the framework unifies differential-form methods with Graham-type integrability conditions for dissipative dynamics. The results suggest that, at least for the studied classes, dissipative systems can be recast as gradient or gradient-like systems via generalized coordinate transformations, with implications for both finite- and infinite-dimensional settings.

Abstract

We find that to the dynamics of a given dissipative system a $p=1$ differential form can be associated with a general decomposition into a potential term and a non-potential residual part. If the residual part is absent the form is closed and the system is gradient system or gradient like. If it is non-closed, in the differential form approach, it remains non-closed under a variable change of coordinates, i.e., the system is not a gradient one or a gradient like in any coordinate system. On the other hand, there are claims that a potential should always exists, i.e., the class of dissipative systems and the the class of gradient systems should coincide. We fix this conundrum by introducing a generalized change of coordinates that aims a transformation to a gradient system or a gradient like system. The condition of being closed in the new coordinates of a certain, through the generalized change of coordinates defined differential form, results in a nonlinear differential equation together with a consistency condition. We give examples of physical systems where an analytical solution for the transformation can be found, and hitherto, the potential, but even when the potential is not accessible analytically, we find that it always exist, and therefore we give in principle an affirmative answer to the defining question of this work. Our findings removes loopholes in the question if a potential may exist but it is not known.