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

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.

Ponderomotive dynamics of waves in quasiperiodically modulated media

“…This work can be expanded in several directions. First, one can extend the theory to dissipative waves [68] and vector waves with polarization effects [39,69], which could be important at Bragg resonances. Second, the theory presented here can be used as a stepping stone to improving the understanding of the modulational instabilities in general wave ensembles.…”

Parametric decay of plasma waves near the upper-hybrid resonance

“…We adopt a variational approach to utilize this link efficiently. We assume dissipation to be negligible for simplicity, but the general method used here is extendable to dissipative waves too [15]. It is also to be noted that a related calculation was proposed recently in Ref.…”

“…Using these, one can consider the dielectric tensor (which is Hermitian in the absence of dissipation [15]) as a pseudodifferential operator; i.e.,ǫ 0 = ǫ 0 (t, x,ω,k), where ǫ 0 is a tensor function. (The prefix "pseudo" indicates that the expansion of ǫ 0 inω andk can contain infinite powers; i.e., although expressed in terms of derivatives, such operator can be essentially nonlocal.)…”

Extending geometrical optics: A Lagrangian theory for vector waves

“…Assessing the importance of polarization effects on waves propagating in strongly magnetized plasma will be discussed in a separate paper. Likewise, the method of including dissipation [26] in the above theory will also be described separately. This appendix summarizes our conventions for the Weyl transform.…”

“…[21], we also introduce, in a unified context and an instructive manner, some of the related advances that were made recently in Refs. [24][25][26]. It is expected that the comprehensive analysis presented in this work will facilitate future practical implementations of the proposed theory, particularly in improving ray-tracing simulations.…”