Lagrangian and Hamiltonian structure of magnetofluid models with gyroviscous-like contributions

Many magnetofluid theories, like ideal MHD and various reduced models, exhibit a noncanonical Hamiltonian structure when expressed in Eulerian variables [1]. Of particular interest are magnetofluid models that systematically include contributions due to finite ion gyro-radii. Building on the work of Ref. [2] we generalize the so-called gyro-map 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.