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 Kähler metric, are compared. The authors discuss how a Kähler metric can arise from a complex structure induced by the Hilbert transform. Hybrid and metriplectic flows which combine Hamiltonian and gradient components are examined. A class of metriplectic systems that is generated by completely antisymmetric triple brackets (trilinear brackets) is described and, for finite-dimensional systems, a Lie algebraic interpretation is given. A variety of explicit examples of the several types of flows are given. It is shown that this geometry describes a number of classical ordinary and partial differential equations of interest and that the different metrics give rise to different kinds of dissipation that occur in applications.
For the entire collection see [Zbl 1278.37005].