Friday, August 31, 2012

1208.6193 (Anthony M. Bloch et al.)

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

Anthony M. Bloch, Philip J. Morrison, Tudor S. Ratiu
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.
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