Euler equations on homogeneous spaces and Virasoro orbits
Euler equations on homogeneous spaces and Virasoro orbits
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DOI：
10.1016/s0001-8708(02)00063-4
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发表时间：
2002-10
影响因子：
1.7
通讯作者：
B. Khesin;G. Misiołek
中科院分区：
文献类型：
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作者：
B. Khesin;G. Misiołek
我们证明了以下三个与各种流体力学近似相关的系统：Korteweg-de Vries方程、Camassa-Holm方程和hunt - saxton方程具有相同的对称群和相似的比哈密顿结构。结果表明，它们的位形空间是Virasoro群，这三个动力系统都可以看作是与该群或适当齐次空间上不同的右不变度量相关的测地线流方程。特别地，我们描述了阿诺德将欧拉方程作为单侧不变度量的测地线流的方法如何从李群扩展到齐次空间。我们还证明了上述三种情况描述了与Virasoro群相关的所有可通过平移论证原理进行积分的一般双哈密顿系统：它们精确地对应于三种不同类型的一般Virasoro轨道。最后，我们讨论了上述度量与Virasoro轨道上的Kahler结构之间的相互关系，以及对应于更精细的轨道分类的可积系统的开放性问题。
We show that the following three systems related to various hydrodynamical approximations: the Korteweg–de Vries equation, the Camassa–Holm equation, and the Hunter–Saxton equation, have the same symmetry group and similar bihamiltonian structures. It turns out that their configuration space is the Virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right-invariant metrics on this group or on appropriate homogeneous spaces. In particular, we describe how Arnold's approach to the Euler equations as geodesic flows of one-sided invariant metrics extends from Lie groups to homogeneous spaces. We also show that the above three cases describe all generic bihamiltonian systems which are related to the Virasoro group and can be integrated by the translation argument principle: they correspond precisely to the three different types of generic Virasoro orbits. Finally, we discuss interrelation between the above metrics and Kahler structures on Virasoro orbits as well as open questions regarding integrable systems corresponding to a finer classification of the orbits.