Varational Equations and Symmetries in the Lagrangian Formalism

TL;DR

This paper analyzes symmetries in higher-order Lagrangian systems using Anderson-Duchamp-Krupka equations, revealing polynomial structures and applying findings to Poincaré and universal invariance cases.

Contribution

It introduces a polynomial structure for second order derivatives in scalar field Lagrangians and refines the understanding of general symmetries in higher-order Lagrangian formalism.

Findings

Polynomial structure in second order derivatives established

Enhanced characterization of symmetries in second order scalar fields

Application to Poincaré and universal invariance cases

Abstract

Symmetries in the Lagrangian formalism of arbitrary order are analysed with the help of the so-called Anderson-Duchamp-Krupka equations. For the case of second order equations and a scalar field we establish a polynomial structure in the second order derivatives. This structure can be used to make more precise the form of a general symmetry. As an illustration we analyse the case of Lagrangian equations with Poincar\'e invariance or with universal invariance.