Approximate conditions admitted by classes of the Lagrangian L=12(−u′2+u2

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-Nordström black hole.

Approximate Conservation Laws of Nonvariational Differential Equations

“…Recently, we have explored the connection between geometry and approximate metrics [12] in the context of heat conduction, and in [13] we proved that approximate symmetries exist if and only if the metric that defines Kinetic energy admits a nontrivial Homothetic algebra. Following this, we established explicit and generalized geometric conditions for a family of Lagrangians that occupy a special place in the literature surrounding perturbation theory [14].…”

“…Proceeding in the same way as described in Example 1, we find the solution of (15) provides that the multipliers of (14), to the specified form of Λ, are…”

“…Recall the nonlinear wave Equation (14) of Example 2. Expanding the dependent variables to the 1st order in ε, we have u = u 0 + εu 1 , and substituting into (14), gives…”

A study of the approximate singular Lagrangian-conditional Noether symmetries and first integrals

“…In a similar way, for any approximate Lagrangian (5) and an approximate Noether generator (8), we derive the first-order approximate part, I 1 as follows:…”

“…This planted the seed that there exists a strong and deep connection between geometry and approximations. This idea was extended to Lagrangians of partial differential equations [4,5,6] and moreover a self-contained approximate Lie symmetry determining method was successfully devised in [7]. In the variational studies, we unveiled new higher-order approximate versions of Noether's theorem.…”