Some variational principles associated with ODEs of maximal symmetry. Part 1: Equations in canonical form

TL;DR

This paper investigates variational and divergence symmetries in linear ODEs with maximal symmetry, providing explicit first integrals and general theorems applicable to equations of various orders and symmetry dimensions.

Contribution

It introduces new theorems and conjectures on symmetries and first integrals for linear ODEs of maximal symmetry, extending to general orders and symmetry algebra dimensions.

Findings

Explicit forms of first integrals for equations with maximal symmetry

Theorems and conjectures on variational symmetries for general order equations

Applicability to linear equations of arbitrary form and symmetry algebra

Abstract

Variational and divergence symmetries are studied in this paper for linear equations of maximal symmetry in canonical form, 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. Some of these results apply to linear equations of a general form and of arbitrary orders or having a symmetry algebra of arbitrary dimension.