Approximate Noether Symmetries and Collineations for Regular Perturbative Lagrangians

TL;DR

This paper explores the relationship between approximate Noether symmetries and collineations in regular perturbative Lagrangians, revealing how symmetries relate to the geometric properties of the underlying manifold.

Contribution

It establishes the connection between approximate Noether symmetries and the Homothetic algebra of the metric in perturbed Lagrangians, extending symmetry analysis to perturbed systems.

Findings

Approximate Noether symmetries are linked to elements of the Homothetic algebra.

Exact symmetries can become approximate symmetries under perturbations.

Applications demonstrate the theoretical results in specific cases.

Abstract

Regular perturbative Lagrangians that admit approximate Noether symmetries and approximate conservation laws are studied. Specifically, we investigate the connection between approximate Noether symmetries and collineations of the underlying manifold. In particular we determine the generic Noether symmetry conditions for the approximate point symmetries and we find that for a class of perturbed Lagrangians, Noether symmetries are related to the elements of the Homothetic algebra of the metric which is defined by the unperturbed Lagrangian. Moreover, we discuss how exact symmetries become approximate symmetries. Finally, some applications are presented.