Mathematics – Symplectic Geometry
Scientific paper
2002-10-25
Advances in Mathematics, vol.176 (2003), 116-144
Mathematics
Symplectic Geometry
26 pages, 4 figures, LaTeX. Advances in Mathematics (to appear)
Scientific paper
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.
Khesin Boris
Misiolek Gerard
No associations
LandOfFree
Euler equations on homogeneous spaces and Virasoro orbits does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Euler equations on homogeneous spaces and Virasoro orbits, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Euler equations on homogeneous spaces and Virasoro orbits will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-614545