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

Given a principal bundle G↪P→B (each being compact, connected and oriented) and a G-invariant metric hP on P which induces a volume form μP, we consider the group of all unimodular automorphisms SAut(P,μP):={φ∈Diff(P)|φ∗μP=μP andφ is G-equivariant} of P, and determines its Euler equation à 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,μP) is an extension of a certain volume preserving diffeomorphisms group of B by the gauge group Gau(P) of P.