The Euler-Poincare equations and double bracket dissipation
The Euler-Poincare equations and double bracket dissipation
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DOI：
10.1007/bf02101622
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发表时间：
1996-01-01
影响因子：
2.4
通讯作者：
Ratiu, TS
中科院分区：
文献类型：
--
作者：
Bloch, A;Krishnaprasad, PS;Ratiu, TS
本文研究了一类Lie-Poisson系统（或Euler-Poincare系统）在具有Brockett双括号形式的特殊耗散项作用下的扰动。我们表明，一个正式的不稳定平衡的未扰动系统成为一个光谱，因此非线性不稳定平衡后，扰动被添加。我们还研究了这种耗散机制的几何形状及其与瑞利耗散函数的关系。这项工作补充了我们早期的工作（布洛赫，Krishnaprasad，Marsden和Ratiu [1991，1994]），其中我们研究了相应的问题，系统的对称性与耗散添加到内部变量;在这里，它是直接添加到组或李代数变量。这里讨论的机制包括一些有趣的例子，如铁磁性的朗道-Lifschitz方程，耗散刚体动力学和地球物理流体的某些模型，以及等离子体物理学和恒星动力学中的某些相对平衡。
This paper studies the perturbation of a Lie-Poisson (or, equivalently an Euler-Poincare) system by a special dissipation term that has Brockett's double bracket form. We show that a formally unstable equilibrium of the unperturbed system becomes a spectrally and hence nonlinearly unstable equilibrium after the perturbation is added. We also investigate the geometry of this dissipation mechanism and its relation to Rayleigh dissipation functions. This work complements our earlier work (Bloch, Krishnaprasad, Marsden and Ratiu [1991, 1994]) in which we studied the corresponding problem for systems with symmetry with the dissipation added to the internal variables; here it is added directly to the group or Lie algebra variables. The mechanisms discussed here include a number of interesting examples of physical interest such as the Landau-Lifschitz equations for ferromagnetism, certain models for dissipative rigid body dynamics and geophysical fluids, and certain relative equilibria in plasma physics and stellar dynamics.