Properties of the Scalar Universal Equations

TL;DR

This paper reviews and generalizes the variational properties of scalar universal equations, revealing new insights into their structure and relation to symmetry algebra, challenging previous assumptions.

Contribution

It introduces a new interpretation of the Euler hierarchy using variational calculus and clarifies its relation to symmetry algebra, correcting earlier claims.

Findings

Each Euler hierarchy member can have explicit field dependence

The Euler hierarchy is linked to the algebra of distinguished symmetries

A new interpretation of the Euler hierarchy in variational calculus context

Abstract

The variational properties of the scalar so--called ``Universal'' equations are reviewed and generalised. In particular, we note that contrary to earlier claims, each member of the Euler hierarchy may have an explicit field dependence. The Euler hierarchy itself is given a new interpretation in terms of the formal complex of variational calculus, and is shown to be related to the algebra of distinguished symmetries of the first source form.