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The ideal magnetohydrodynamics (MHD) as well as the ideal fluid dynamics is governed by the Hamiltonian equation with respect to the Lie-Poisson bracket. The Nambu bracket manifestly represents the Lie-Poisson structure in terms of derivative of the Casimir invariants. We construct a compact Nambu-bracket representation for the three-dimensional ideal MHD equations, with use of three Casimirs for the second Hamiltonians, the total entropy and the magnetic and cross helicities, whose coefficients are all constant. The Lie-Poisson bracket induced by this Nambu bracket does not coincide with the original one, but supplemented by terms with an auxiliary variable. The supplemented Lie-Poisson bracket qualifies the cross-helicity as the Casimir.
A Casimir invariant is a constant integral for arbitrary functional form of the Hamiltonian and is regarded as a topological invariant. By appealing to Noether’s theorem in the variational framework, we show that the cross-helicity is the integral invariant associated with the particle-relabeling symmetry. By incorporating the divergence symmetry, all the other known topological invariants are put on the same ground of Noether’s theorem. This is a sort of gauge theory.