Some variational principles associated with ODEs of maximal symmetry. Part 2: The general case

TL;DR

This paper explores variational and divergence symmetries in linear and nonlinear ODEs of maximal symmetry, providing explicit first integrals and analyzing the impact of different Lagrangians on symmetry properties.

Contribution

It extends the analysis of variational symmetries to a broad class of maximal symmetry ODEs, including general order equations, and compares different Lagrangian formulations.

Findings

Explicit first integrals for equations of maximal symmetry.

Differences in symmetry results based on Lagrangian choice.

General theorems and conjectures for equations of arbitrary order.

Abstract

Variational and divergence symmetries are studied in this paper for the whole class of linear and nonlinear equations of maximal symmetry, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. A discussion of the existence of variational symmetries with respect to a different Lagrangian, which turns out to be the most common and most readily available one, is also carried out. This leads to significantly different results when compared with the former case of the transformed Lagrangian. The latter analysis also gives rise to more general results concerning the variational symmetry algebra of any linear or nonlinear equations