Direct approach to approximate conservation laws
M. Gorgone, G. Inferrera
TL;DR
The paper develops a perturbation-consistent extension of the direct multiplier method to derive approximate conservation laws for non-variational differential equations with a small parameter $\varepsilon$. It introduces approximate multipliers dependent on $\varepsilon$ and defines an approximate Euler operator to enforce divergence conditions, proving a theorem that guarantees when a set of multipliers yields an approximate conservation law. The authors compare approaches from the literature and demonstrate the method on several non-variational models, obtaining first-order approximate conserved quantities that align with perturbation theory and that are computable with computer algebra. This framework provides a cost-effective tool for identifying conserved structures in perturbed, non-variational systems and sets the stage for further connections with approximate Noether theory and self-adjoint formulations.
Abstract
In this paper, non-variational systems of differential equations containing small terms are considered, and a consistent approach for deriving approximate conservation laws through the introduction of approximate Lagrange multipliers is developed. The proposed formulation of the approximate direct method starts by assuming the Lagrange multipliers to be dependent on the small parameter; then, by expanding the dependent variables in power series of the small parameter, we consider the consistent expansion of all the involved quantities (equations and Lagrange multipliers) in such a way the basic principles of perturbation analysis are not violated. Consequently, a theorem leading to the determination of approximate multipliers whence approximate conservation laws arise is proved, and the role of approximate Euler operators emphasized. Some applications of the procedure are presented.