Hamiltonian structure of magnetofluid models with gyroviscous-like contributions

Magnetofluid theories, like MHD, can be expressed in terms of Eulerian (or spatial) variables, or in terms of Lagrangian (or material) variables. The former formulation generally exhibits a noncanonical Hamiltonian structure [1]. Building on the work of Ref. [2] we generalize the gyromap to three dimensional magnetofluid theories. Starting with the 3D ideal MHD noncanonical Poisson bracket [1] and a Hamiltonian including general gyroviscous terms, we derive equations of motions and compare them to, e.g., Braginskii [3] in the collisionless limit. In addition we explore the Lagrangian version of these theories which use Hamilton's principle to derive the equations of motion [4]. \\[4pt] [1] P.J.~Morrison and J.M.~Greene, Phys. Rev. A {\bf 45},790 (1980).\\[0pt] [2] P.J.~Morrison, I.L.~Caldas, and H.~Tasso, Z. Naturforsch. {\bf 39a}, 1023 (1984).\\[0pt] [3] S.I.~Braginskii, in {\it Review of Plasma Physics}, ed. M.A.~Leontovich (Consultants Bureau, New York, 1965), Vol. 1, p. 205.\\[0pt] [4] W.A.~Newcomb, Nuclear Fusion: 1962 Suppl. Part 2, p. 451.