An effective potential for classical Yang-Mills fields as outline for bifurcation on gauge orbit space

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Classical Yang-Mills (Y.M.) equations with static external sources are formulated as a Hamiltonian system with gauge symmetry in A 0 = 0 gauge. Using the concept of a “momentum mapping” (J. Marsden and A. Weinstein, Rep. Math. Phys. 5 (1974), 121) on symplectic manifolds with symmetry, an analogue of centrifugal potential of a mass point in a spherically symmetric potential is derived. This gives rise to an effective potential V eff, whose critical points are rigorously proved to be in one-to-one correspondence with static Y.M. solutions. V eff additionally depends on the prescribed external source ϱ, which is as a constant of motion analogous to angular moment of the mass point. Thus bifurcation of static solutions is caused by bifurcation of critical points of V eff under variation of the external parameter ϱ. Some closing remarks on dynamics and stability on gauge orbit space are added.