Approximate conditions admitted by classes of the Lagrangian ${\cal L}=\frac12\left(-u'^2+u^2\right)+\epsilon^iG_i(u, u^\prime, u^{\prime\prime})$

TL;DR

This paper develops explicit geometric conditions for approximate symmetries in a class of perturbed harmonic oscillator Lagrangians, enabling systematic analysis of approximate solutions in various physical contexts.

Contribution

It introduces a generalized geometric framework for identifying approximate symmetries in perturbed Lagrangian systems, applicable to diverse differential equations.

Findings

Derived explicit symmetry conditions for perturbed harmonic oscillators.

Applied conditions to specific cases like quadratic G_1, Klein-Gordon, and black hole orbital equations.

Generated nontrivial approximate symmetries and transformations systematically.

Abstract

We investigate a class of Lagrangians that admit a type of perturbed harmonic oscillator which occupies a special place in the literature surrounding perturbation theory. We establish explicit and generalized geometric conditions for the symmetry determining equations. The explicit scheme provided can be followed and specialized for any concrete perturbed differential equation possessing the Lagrangian. A systematic solution of the conditions generate nontrivial approximate symmetries and transformations. Detailed cases are discussed to illustrate the relevance of the conditions, namely (a) $G_1$ as a quadratic polynomial, (b) the Klein-Gordon equation of a particle in the context of Generalized Uncertainty Principle and (c) an orbital equation from an embedded Reissner-Nordstr\"om black hole.