Gradient flows in the normal and K\"ahler metrics and triple bracket   generated metriplectic systems
  The dynamics of gradient and Hamiltonian flows with particular application to flows on adjoint orbits of a Lie group and the extension of this setting to flows on a loop group are discussed. Different types of gradient flows that arise from different metrics including the so-called normal metric on adjoint orbits of a Lie group and the K\"ahler metric are compared. It is discussed how a K\"ahler metric can arise from a complex structure induced by the Hilbert transform. Hybrid and metriplectic flows that arise when one has both Hamiltonian and gradient components are examined. A class of metriplectic systems that is generated by completely antisymmetric triple brackets is described and for finite-dimensional systems given a Lie algebraic interpretation. A variety of explicit examples of the several types of flows are given. 
Anthony M. Bloch, Philip J. Morrison, and Tudor S. Ratiu
math-ph math.DS math.MP physics.flu-dyn physics.plasm-ph