  In this paper we investigate the use of the vector potential as a means of
maintaining the divergence constraint in the numerical solution of the
equations of Magnetohydrodynamics (MHD) using the Smoothed Particle
Hydrodynamics (SPH) method. We derive a self-consistent formulation of the
equations of motion using a variational principle that is constrained by the
numerical formulation of both the induction equation and the curl operator used
to obtain the magnetic field, which guarantees exact and simultaneous
conservation of momentum, energy and entropy in the numerical scheme. This
leads to a novel formulation of the MHD force term, unique to the vector
potential, which differs from previous formulations. We also demonstrate how
dissipative terms can be correctly formulated for the vector potential such
that the contribution to the entropy is positive definite and the total energy
is conserved.
  On a standard suite of numerical tests in one, two and three dimensions we
find firstly that the consistent formulation of the vector potential equations
is unstable to the well-known SPH tensile instability, even more so than in the
standard Smoothed Particle Magnetohydrodynamics (SPMHD) formulation where the
magnetic field is evolved directly. Furthermore we find that, whilst a hybrid
approach based on the vector potential evolution equation coupled with a
standard force term gives good results for one and two dimensional problems
(where dAz/dt = 0), such an approach suffers from numerical instability in
three dimensions related to the unconstrained evolution of vector potential
components. We conclude that use of the vector potential is not a viable
approach for Smoothed Particle Magnetohydrodynamics.
