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.

中文翻译：

---

非线性静磁变分原理及其对应关系

我们使用涉及机械变形和代表磁性分量的独立场的几种变分公式推导出非线性磁弹性静力学方程。还讨论了使用勒让德变换和麦克斯韦方程组特性的这些公式之间的等价性，以某些边界积分或常数积分为模。对于增量场以及所有五个变分原理，都说明了基于第二个变分的分岔方程。当在身体周围的无限空间上定义总势能时，我们发现能量原理中包含某些与外加磁场相关的术语会导致麦克斯韦应力张量和相关边界条件的轻微变化。另一方面，当磁场中包含的能量限于有限体积时，我们发现所讨论的公式与物理实体的相关表达式之间存在对应关系。鉴于文献中研究的问题中的一组不同的边界数据和外部应用控制的性质，以及建模中采用的同样不同的变分原理列表，我们的分析强调在选择变分原理和未知领域时要谨慎，因此与其他选择的一致性也得到满足。