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arXiv · 1208.6193

Gradient flows in the normal and Kähler metrics and triple bracket generated metriplectic systems

Anthony M. Bloch·Philip J. Morrison·Tudor S. Ratiu
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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 Kähler metric are compared. It is discussed how a Kähler 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.

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Anthony M. Bloch, Philip J. Morrison, Tudor S. Ratiu. 2012-08-30. Gradient flows in the normal and Kähler metrics and triple bracket generated metriplectic systems. https://arxiv.org/abs/1208.6193

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