  We derive the equations of nonlinear magnetoelastostatics using several
variational formulations involving the mechanical deformation and an
independent field representing the magnetic component. An equivalence is also
discussed, modulo certain boundary integrals or constant integrals, between
these formulations using the Legendre transform and properties of Maxwell's
equations. The second variation based bifurcation equations are stated for the
incremental fields as well for all five variational principles. When the total
potential energy is defined over the infinite space surrounding the body, we
find that the inclusion of certain term in the energy principle, associated
with the externally applied magnetic field, leads to slight changes in the
Maxwell stress tensor and associated boundary conditions. On the other hand,
when the energy contained in the magnetic field is restricted to finite
volumes, we find that there is a correspondence between the discussed
formulations and associated expressions of physical entities. In view of a
diverse set of boundary data and nature of externally applied controls in the
problems studied in the literature, along with a equally diverse list of
variational principles employed in modeling, our analysis emphasizes care in
the choice of variational principle and unknown fields so that consistency with
other choices is also satisfied.
