Hilbert–Yang–Mills functional: Examples

Abstract

The geometric structure of gauge natural theories is investigated. We study especially the Einstein–Yang–Mills theory. We discuss the first variation formula by means of the principal Lepage equivalent of the Hilbert–Yang–Mills Lagrangian. We establish basic invariant properties and the first variation formula for induced variations of this Lagrangian. Noether currents split naturally to several terms, one of which is the exterior derivative of the Komar–Yang–Mills superpotential. We give some examples of Komar–Yang–Mills superpotentials corresponding to solutions of the Einstein equations.