This chapter describes the Hamiltonian approach to MHD and gas dynamics. In Sect. 8.1 we describe a constrained MHD variational principle by using Lagrange multipliers to enforce the constraints of mass conservation; the entropy advection equation; Faraday’s equation and the so-called Lin constraint describing in part, the vorticity of the flow (i.e. Kelvin’s theorem). This leads to Hamilton’s canonical equations in terms of Clebsch potentials. The Lagrange multipliers define the Clebsch variables, which gives a Clebsch representation for the fluid velocity u (Zakharov and Kuznetsov (1997)). In Sect. 8.2 we transform the canonical, Clebsch variable, Poisson bracket to different non-canonical forms that use Eulerian physical variables (see e.g. Morrison and Greene (1980, 1982), Morrison (1982)) and Holm and Kupershmidt (1983a,b)). The different MHD brackets are described in Sect. 8.3. Section 8.4 verifies the Jacobi identity for the bracket of Morrison and Greene (1982) in which ∇⋅B can be non-zero. We discuss how the Morrison and Greene (1980) bracket, with ∇⋅B = 0, has been placed on a more rigorous footing by the use of the Dirac bracket and projectors by Chandre et al. (2012); Chandre (2013); Chandre et al. (2013) (see also Banerjee and Kumar (2016)). We use the functional multi-vector approach of Olver (1993) to investigate and check the Jacobi identity for: (i) the Morrison and Greene (1982) bracket (ii) the advected A bracket in which A ⋅ dx is Lie dragged with the flow used by (Holm and Kupershmidt (1983a,b)) and (iii) the Morrison and Greene (1980) bracket. The non-canonical Poisson brackets are used to determine the MHD Casimirs in Sect. 8.5 (e.g. Hameiri (2004)). The Casimirs are related to the advected invariants.
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