Variational principles for dissipative (sub)systems, with applications to the theory of linear dispersion and geometrical optics

TL;DR

This paper introduces a novel variational approach for dissipative systems using nonlocal Lagrangians, enabling unified treatment of reversible and irreversible dynamics, with applications to linear dispersion and geometrical optics in complex media.

Contribution

It proposes a new variational framework for dissipative systems using constant Lagrange multipliers and nonlocal Lagrangians, extending variational principles to non-conservative systems.

Findings

Formulated a variational principle for linear dispersion.

Developed a variational formulation for linear geometrical optics in complex media.

Demonstrated the approach's applicability to nonstationary, inhomogeneous, and anisotropic media.

Abstract

Applications of variational methods are typically restricted to conservative systems. Some extensions to dissipative systems have been reported too but require ad hoc techniques such as the artificial doubling of the dynamical variables. Here, a different approach is proposed. We show that, for a broad class of dissipative systems of practical interest, variational principles can be formulated using constant Lagrange multipliers and Lagrangians nonlocal in time, which allow treating reversible and irreversible dynamics on the same footing. A general variational theory of linear dispersion is formulated as an example. In particular, we present a variational formulation for linear geometrical optics in a general dissipative medium, which is allowed to be nonstationary, inhomogeneous, nonisotropic, and exhibit both temporal and spatial dispersion simultaneously.