On the single field formulation in magnetostatics
Pith reviewed 2026-05-20 06:56 UTC · model grok-4.3
The pith
The single-field formulation using only magnetic induction is equivalent to the magnetization and field formulation in magnetostatics, and this equivalence stays intact when the models are coupled to elasticity.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The equivalence of the magnetization-field formulation and the single magnetic induction formulation of magnetostatics is stable under coupling with elasticity. The corresponding magnetoelastic energy densities are obtained via Legendre-Fenchel transform in the magnetic state variables, but the two overall functionals are not linked by standard convex duality. Convexity and coercivity of the functional are neither required for the transformation nor always preserved by it.
What carries the argument
The transformation that converts between the two-field magnetization formulation and the single induction formulation, performed through the Legendre-Fenchel transform applied to the magnetic state variables.
If this is right
- The single induction formulation can be used interchangeably with the two-field version in magnetoelastic problems.
- Equivalence between the formulations does not rely on convexity or coercivity assumptions.
- The link between the formulations remains available for any static model coupled to elasticity.
- One formulation can be substituted for the other even when the energy densities are related only by the Legendre-Fenchel transform.
Where Pith is reading between the lines
- Numerical schemes for magnetoelastic materials could freely select whichever formulation is easier to discretize.
- Similar equivalences without standard duality might exist when magnetostatics is coupled to other mechanical or thermal models.
- The result suggests that variational reformulations in coupled systems can be justified under weaker analytic conditions than convexity alone.
Load-bearing premise
The magnetoelastic energy densities can be recovered from the Legendre-Fenchel transform in the magnetic variables while the complete functionals still remain equivalent without ordinary convex duality at the functional level.
What would settle it
A concrete magnetoelastic boundary-value problem in which the two functionals attain different minimum values or produce distinct minimizers would show the claimed equivalence does not hold.
read the original abstract
We systematically discuss the equivalence of two variational formulations of magnetostatics, in terms of magnetization and magnetic field on the one hand and the single field formulation using only magnetic induction. To demonstrate that this link is stable also when the magnetic laws are coupled with other variational static models, elasticity is included in the models as well. Interestingly, despite the fact that the corresponding magnetoelastic energy densities in the material can be computed via Legendre-Fenchel transform in the magnetic state variables, the two formulations are not linked by standard convex duality on the level of the functionals. In addition, convexity and coercivity of the given functional are neither required for the transformation nor always preserved by it.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper systematically discusses the equivalence of two variational formulations of magnetostatics: one involving magnetization and magnetic field, and a single-field formulation using only magnetic induction. It extends the analysis to include coupling with elasticity to demonstrate stability of the equivalence. The central claims are that magnetoelastic energy densities obtained via Legendre-Fenchel transform in the magnetic variables yield equivalent functionals, that the formulations are not linked by standard convex duality at the functional level, and that convexity and coercivity are neither required for nor always preserved by the transformation.
Significance. If the equivalence proofs and stability under magnetoelastic coupling hold rigorously, the work offers a practical reduction to a single-field variational model for magnetostatics and coupled systems, which could simplify analysis and computation in materials modeling. The distinction between density-level Legendre-Fenchel transforms and the absence of standard convex duality at the functional level is a noteworthy theoretical observation, as is the finding that convexity/coercivity need not be preserved.
major comments (2)
- [Abstract] Abstract and the magnetoelastic coupling discussion: the claim that equivalence of the two formulations remains stable under coupling to elasticity requires explicit total-energy expressions and the precise function-space setting to verify that the elastic term (which may depend on the deformation gradient induced by magnetic body forces or magnetostriction) commutes with the Legendre-Fenchel transform while preserving the set of minimizers. Without these, it is unclear whether the equivalence of critical points holds when the elastic energy depends on the magnetic state.
- [Magnetoelastic coupling section] The assertion that the formulations are not linked by standard convex duality on the level of the functionals, despite using Legendre-Fenchel transforms on the densities, needs a concrete counter-example or explicit functional comparison showing where duality fails at the global level while holding locally.
minor comments (2)
- Clarify the precise functional setting (e.g., Sobolev spaces or admissible field classes) used for the equivalence proof, as this is essential for assessing coercivity and existence of minimizers.
- Add a brief remark on how the single-field formulation handles the divergence-free constraint on the magnetic induction explicitly.
Simulated Author's Rebuttal
We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and will incorporate clarifications and additions in the revised version.
read point-by-point responses
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Referee: [Abstract] Abstract and the magnetoelastic coupling discussion: the claim that equivalence of the two formulations remains stable under coupling to elasticity requires explicit total-energy expressions and the precise function-space setting to verify that the elastic term (which may depend on the deformation gradient induced by magnetic body forces or magnetostriction) commutes with the Legendre-Fenchel transform while preserving the set of minimizers. Without these, it is unclear whether the equivalence of critical points holds when the elastic energy depends on the magnetic state.
Authors: We agree that additional explicit details would improve clarity. In the revised manuscript we will state the total magnetoelastic energy explicitly as the sum of the magnetic part (subject to the Legendre-Fenchel transform in the induction variable) and the elastic part (depending on the deformation gradient). We will specify the function spaces (e.g., H(curl) for the magnetic induction and W^{1,p} for the deformation) and show that the transform acts only on the pointwise magnetic density. Because the elastic energy enters through a variational coupling that is independent of the choice of magnetic variable, the sets of critical points remain in one-to-one correspondence; the magnetic body-force and magnetostriction contributions transform consistently under the duality map. These additions will be placed in a new subsection of the magnetoelastic coupling section. revision: yes
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Referee: [Magnetoelastic coupling section] The assertion that the formulations are not linked by standard convex duality on the level of the functionals, despite using Legendre-Fenchel transforms on the densities, needs a concrete counter-example or explicit functional comparison showing where duality fails at the global level while holding locally.
Authors: We accept that a concrete illustration is desirable. In the revision we will insert a short subsection containing an explicit functional counter-example: a simple quadratic density whose Legendre-Fenchel transform yields an equivalent local problem, yet the associated integral functionals fail to be dual in the classical convex-analysis sense because of the divergence-free constraint on the induction field and the boundary conditions. The example will demonstrate that inf-sup interchange does not hold globally even though it holds pointwise, thereby justifying the distinction drawn in the manuscript. revision: yes
Circularity Check
No significant circularity; equivalence shown via direct variational arguments independent of self-referential definitions or fitted inputs
full rationale
The paper presents a systematic discussion and proof of equivalence between two variational formulations of magnetostatics, extended to include elasticity. The abstract and context indicate that the link is established through explicit transformation of energy densities using the Legendre-Fenchel transform in magnetic variables, with the overall functionals shown to share minimizers despite not being related by standard convex duality. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central claim rests on a direct equivalence argument in the function-space setting. The inclusion of elasticity is handled by verifying that the transform preserves the set of critical points without requiring convexity/coercivity preservation. This is a self-contained mathematical derivation against standard benchmarks in variational calculus, warranting a score of 0 with no circular steps identified.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The magnetoelastic energy densities admit a Legendre-Fenchel transform with respect to the magnetic state variables.
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel echoes?
echoesECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.
the corresponding magnetoelastic energy densities in the material can be computed via Legendre-Fenchel transform in the magnetic state variables, the two formulations are not linked by standard convex duality on the level of the functionals
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IndisputableMonolith/Foundation/AlphaCoordinateFixation.leanJ_uniquely_calibrated_via_higher_derivative unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Equivalence statement for the smooth convex case... Φ(x,F,b) = −bΨ∗(x,F,b)
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
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discussion (0)
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