We prove that any regular Casimir in 3D magnetohydrodynamics is a function of the magnetic helicity and cross-helicity. In other words, these two helicities are the only independent regular integral invariants of the coadjoint action of the MHD group $\text{SDiff}(M)\ltimes\mathfrak X^*(M)$, which is the semidirect product of the group of volume-preserving diffeomorphisms and the dual space of its Lie algebra.

中文翻译：

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Casimirs 在 3D 磁流体动力学中的基础

我们证明 3D 磁流体动力学中的任何规则 Casimir 都是磁螺旋度和交叉螺旋度的函数。换句话说，这两个螺旋是 MHD 群 $\text{SDiff}(M)\ltimes\mathfrak X^*(M)$ 的协同作用的唯一独立正则积分不变量，它是群保体积微分同胚及其李代数的对偶空间。