Further
ANNUAL REVIEWSis < [u] ,v) = SMU(V)J.l where VEt § and the form UEnl is an arbitrary representative of [u] E nl/dno• Let G = SDiff(M) be the group of all volume-preserving diffeo morphisms. Then the defi nition of the change of variables in the integral and the invariance of J.l imply the coincidence of the coadjoint G-action with the G-action on the space of I-forms.The crucial point of the proof is the fo llowingProposition The integrals I(u) and IJ(u) given by (2a,b) are well defined fu nctionals on ' §* (i.e. they don't depend on the choice of the representative U in the class [u J) and are invariants of the coadjoint action.Proof Since the coadjoint action is push-forward, the statement follows from the change of variables formula and the coordinate-free defi nition of the corresponding integrals.Let (. , .) be a Riemannian metric on M (whose volume form differs, in general, from the given volume J.l). It defines a nondegenerate scalar pro- Annu. Rev. Fluid Mech. 1992.24:145-166. Downloaded from www.annualreviews.org by Indiana University -Purdue University Indianapolis -IUPUI on 10/11/12. For personal use only.
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.
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