A Basis of Casimirs in 3D Magnetohydrodynamics
Boris Khesin, Daniel Peralta‐Salas, Cheng Yang
Abstract
Abstract We prove that any regular Casimir in 3D magnetohydrodynamics (MHD) 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 $\textrm{SDiff}(M)\ltimes \mathfrak X^*(M)$, which is the semidirect product of the group of volume-preserving diffeomorphisms and the dual space of its Lie algebra.
Topics & Concepts
Magnetic helicitySemidirect productMagnetohydrodynamicsCasimir effectMathematicsHelicityGroup (periodic table)Action (physics)Pure mathematicsSpace (punctuation)Basis (linear algebra)Mathematical physicsDual spaceProduct (mathematics)Lie algebraAlgebra over a fieldPhysicsMagnetic fieldClassical mechanicsGeometryQuantum mechanicsComputer scienceOperating systemBlack Holes and Theoretical PhysicsQuantum chaos and dynamical systemsCosmology and Gravitation Theories