Gradient Flows in the Normal and Kähler Metrics and Triple Bracket Generated Metriplectic Systems
Gradient Flows in the Normal and Kähler Metrics and Triple Bracket Generated Metriplectic Systems
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普通和 Kähler 度量以及三重括号生成的 Metriplectic 系统中的梯度流
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
T. Ratiu
中科院分区：
文献类型：
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作者：
A. Bloch;P. Morrison;T. Ratiu
讨论了梯度流和哈密顿流的动力学问题，特别是在李群伴随轨道上的流动中的应用，并将这一设置推广到环群上的流动。本文比较了由李群伴随轨道上的所谓法向度规和Kahler度规等不同度量产生的不同类型的梯度流。讨论了如何从希尔伯特变换引起的复杂结构中产生卡勒度规。研究了结合哈密顿分量和梯度分量的混合流和三元流。描述了由完全反对称三方括号（三线性方括号）生成的一类三重系统，并给出了有限维系统的李代数解释。给出了这几种流的各种显式例子。结果表明，这种几何结构描述了许多经典的常微分方程和偏微分方程，不同的度量会引起应用中出现的不同类型的耗散。
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.