Gradient Flows in the Normal and Kähler Metrics and Triple Bracket Generated Metriplectic Systems

Abstract

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 Kahler metric are compared. It is discussed how a Kahler 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 given a Lie algebraic interpretation. 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.