## JMI2009B-7 A comprehensive view of Lagrangian invariants of hydrodynamics, ideal and Hall magnetohydrodynamics on three-dimensional Riemannian manifold (pp.139-147)

Author(s)： Keisuke Araki

J. Math-for-Ind. 1B (2009) 139-147.

Abstract
Lagrangian invariants of hydrodynamic, magnetohydrodynamic (MHD) and Hall MHD fluids are reviewed in a general viewpoint of differential topology. It is shown that, introducing the particle trajectory map (PTM) and its inverse (back-to-labels map, BLM) and utilizing their spatial derivatives, one can easily derive the conservation laws along the Lagrangian trajectories. All the invariants are derived as composite of such elementary invariants as entropy per unit mass, impulse, mass density, and electromagnetic vector potential and their derivatives. Treating the spatial derivatives of PTM and BLM as kinds of Lagrangian invariants formally, one can understand the following conservation laws as Lagrangian invariants:Cauchy's formula, Weber's transformation, Ertel's theorem, Ertel-Rossby's theorem (i.e. helicity density), magnetic-helicity and cross-helicity in a MHD fluid, hybrid-helicity in a Hall MHD fluid.

Keyword(s).　 Lagrangian invariants, conservation laws, diﬀerential topology, barotropic fluids