Nambu brackets and induced Lie-Poisson brackets for fluid mechanics and magnetohydrodynamics

For the ideal magnetohydrodynamics (MHD), Noether's theorem states that the topological invariant associated with the particle relabeling symmetry is the cross helicity, the volume integral of the scalar product of the velocity field and a frozen-in field. This is also the case for the dynamics of an ideal fluid. A proof to it is given in terms of variation of the Lagrangian label as a function of the Eulerian position. In addition to the cross helicity, the total mass, the total entropy and the magnetic helicity are topological invariants. We construct the Nambu bracket for the ideal MHD, using all the four topological invariants as Hamiltonians, together with the total energy. The Lie-Poisson bracket induced from the Nambu bracket gives an extension of the known one and automatically guarantees the cross-helicity to be a Casimir invariant. A remark is given to Noether's second theorem.