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

Abstract

The investigation of approximate symmetries of reparametrization invariant Lagrangians of n + 1 degrees of freedom and quadratic velocities is presented. We show that extra conditions emerge which give rise to approximate and conditional Noether symmetries of such constrained actions. The Noether symmetries are the simultaneous conformal Killing vectors of both the kinetic metric and the potential. In order to recover these conditional symmetry generators which would otherwise be lost in gauge fixing the lapse function entering the perturbative Lagrangian, one must consider the lapse among the degrees of freedom. We establish a geometric framework in full generality to determine the admitted Noether symmetries. Additionally, we obtain the corresponding first integrals (modulo a constraint equation). For completeness, we present a pedagogical application of our method.

Imaging Noise Suppression: Fourth-Order Partial Differential Equations and Travelling Wave Solutions

“…We consider this noise removal model within the framework of Lie symmetries, mainly to obtain exact solutions to the model. The immense and diversified contribution of symmetries to all areas of science can be seen in, among numerous others, the works of [10][11][12][13][14][15][16]. An important byproduct of the Lie point symmetry method is that one may use its applications to recognise patterns in imaging [17,18].…”