On the Single Field Formulation in Magnetostatics
Stefan Krömer and Giuseppe Tomassetti (2026)
Accepted for publication in Mathematics and Mechanics of Solids.
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Motivation
Magnetostatic problems in magnetic and magnetoelastic materials are commonly written in two different sets of variables. In one formulation, the magnetic state is described by the magnetization (\mathbf m) together with the magnetic field (\mathbf h). An alternative, single-field formulation uses only the magnetic induction (\mathbf b), subject to the solenoidal Maxwell constraint.
The two descriptions refer to the same physical magnetic state through
\[\mathbf b = \mu_0(\mathbf m + \mathbf h),\]but this identity does not by itself make their variational formulations equivalent. The corresponding energies are defined on different fields, obey different constraints, and can have different mathematical properties. Understanding their precise relationship matters when deriving constitutive models, proving the existence of equilibria, or choosing variables for numerical computation.
Two descriptions of the magnetic state
The ((\mathbf m,\mathbf h))-based formulation separates the material response, represented by the magnetization, from the magnetic field. The single-field formulation uses the Eulerian magnetic induction (\mathbf b) as its basic unknown. In magnetoelasticity, the induction can alternatively be pulled back to the reference configuration and represented by the Lagrangian field (\mathbf B); the paper focuses primarily on the Eulerian induction (\mathbf b).
Although the material energy densities can be related by Legendre–Fenchel transformations in the magnetic variables, this local relation should not be confused with an ordinary convex-duality relation between the complete energy functionals.
Main result
The paper establishes, under the appropriate hypotheses, a two-way correspondence between constrained equilibrium states. A critical point of the ((\mathbf m,\mathbf h)) formulation determines a critical point of the (\mathbf b) formulation, and conversely. The result is proved both for smooth constitutive laws and, using subdifferentials and superdifferentials, for nonsmooth convex or concave laws.
At corresponding equilibrium states, the two formulations assign the same total energy. This is an equilibrium correspondence rather than an assertion that the two functionals agree on every admissible magnetic state.
Why the distinction matters
The distinction has direct analytical and computational consequences. Once its divergence-free constraint is handled, the (\mathbf b)-based formulation has a local structure that can be expressed using a vector potential. By contrast, the self field generated by a prescribed magnetization is intrinsically nonlocal in the ((\mathbf m,\mathbf h)) formulation.
Moreover, properties important for variational analysis need not survive the change of variables. Convexity and coercivity can change between the two formulations; in particular, natural models of diamagnetic response can lead to a concave energy in the ((\mathbf m,\mathbf h)) variables, so that equilibrium is characterized by maximization rather than minimization. These differences matter when proving existence, constructing relaxed theories, and choosing variables for numerical computation.
Magnetoelasticity
Although the central question is magnetostatic, the paper includes elastic deformation in the formulation. The deformation acts as a parameter in the equivalence argument, showing that the relation between the two magnetic descriptions survives coupling to an elastic variational model. This makes the result directly relevant to magnetoelasticity and provides a mathematical foundation for choosing magnetic variables in future models of soft magnetic materials and structures.
Publication
Stefan Krömer and Giuseppe Tomassetti, On the single field formulation in magnetostatics, Mathematics and Mechanics of Solids, accepted for publication, 2026.
The arXiv preprint is available while the journal publication is in preparation.