Abstract
We provide a novel action principle for nonrelativistic ideal magnetohydrodynamics in the Eulerian scheme exploiting a Clebsch-type parametrisation. Both Lagrangian and Hamiltonian formulations have been considered. Within the Hamiltonian framework, two complementary approaches have been discussed using Dirac's constraint analysis. In one case the Hamiltonian is canonical involving only physical variables but the brackets have a noncanonical structure, while the other retains the canonical structure of brackets by enlarging the phase space. The special case of incompressible magnetohydrodynamics is also considered where, again, both the approaches are discussed in the Hamiltonian framework. The conservation of the stress tensor reveals interesting aspects of the theory.
Highlights
Understanding a system from an action principle is always desirable as it offers further insights
The form of the Lagrangian is not unique but varies from author to author, who have employed different approaches, and in the number of basic fields in the Lagrangian. The roots of these ambiguities lie in fluid dynamics itself [9,10]
In this paper we present another approach to obtain the noncanonical brackets starting from an action principle and following Dirac’s constraint analysis [17]
Summary
Understanding a system from an action principle is always desirable as it offers further insights. The MHD Lagrangian proposed in [6] uses density, entropy density, velocity, magnetic field and a new field subject to a constraint, introduced by Lin [21], as the basic fields. The Gauss’ law for magnetism is not incorporated in the Lagrangian itself but is later used to correctly reproduce the force-balance equation, commonly known as the Euler equation In this approach, the physical significance of Lin’s constraint remained obscure. A new approach to obtain the Lagrangian for nonrelativistic perfect fluids, based on Noether’s definition of energymomentum tensor, was advocated in [22], a paper involving one of us This approach naturally dictates a Clebsch-type parametrisation of velocity. The Hamiltonian has the standard canonical structure from which the MHD equations are reproduced by using the Dirac (noncanonical) brackets.
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