A gauge-theoretic description of $\mu$-prolongations, and $\mu$-symmetries of differential equations

TL;DR

This paper introduces a gauge-theoretic framework for understanding $mbda$-prolongations and $mbda$-symmetries in differential equations, extending the concept to non-variational and non-covariant contexts with detailed examples.

Contribution

It provides a gauge-theoretic interpretation of $mbda$-prolongations and $mbda$-symmetries, generalizing previous notions and applying them to broader classes of differential equations.

Findings

$mbda$-prolongations naturally arise from gauge transformations.

The framework applies to non-variational, non-covariant differential equations.

Several detailed examples illustrate the theoretical developments.

Abstract

We consider generalized (possibly depending on fields as well as on space-time variables) gauge transformations and gauge symmetries in the context of general -- that is, possibly non variational nor covariant -- differential equations. In this case the relevant principal bundle admits the first jet bundle (of the phase manifold) as an associated bundle, at difference with standard Yang-Mills theories. We also show how in this context the recently introduced operation of $\mu$-prolongation of vector fields (which generalizes the $\la$-prolongation of Muriel and Romero), and hence $\mu$-symmetries of differential equations, arise naturally. This is turn suggests several directions for further development. S0ome detailed examples are also given.