The group of unimodular automorphisms of a principal bundle and the Euler-Yang-Mills equations

Given a principal bundle G hooked right arrow P -> B (each being compact, connected and oriented) and a G-invariant metric h(P) on P which induces a volume form mu(P), we consider the group of all unimodular automorphisms SAut(P, mu(P)) := {phi is an element of Diff(P) vertical bar phi*mu(P) = mu(P) and phi is G-equivariant) of P, and determines its Euler equation a la Arnold. The resulting equations turn out to be (a particular case of) the Euler-Yang-Mills equations of an incompressible classical charged ideal fluid moving on B. It is also shown that the group SAut(P, mu(P)) is an extension of a certain volume preserving diffeomorphisms group of B by the gauge group Gau(P) of P. (C) 2010 Elsevier B.V. All rights reserved.