Direct approach to approximate conservation laws

TL;DR

This paper introduces a new method for deriving approximate conservation laws in non-variational differential systems with small terms, using perturbation expansions of Lagrange multipliers to ensure consistency.

Contribution

It develops a systematic approach for obtaining approximate conservation laws via an expansion of Lagrange multipliers dependent on a small parameter, ensuring perturbation consistency.

Findings

The method successfully derives approximate conservation laws for specific systems.

The approach maintains the fundamental principles of perturbation analysis.

Applications demonstrate the effectiveness of the proposed procedure.

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.