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

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.