The Camassa-Holm equation as a geodesic flow on the diffeomorphism group

Misiolek [J. Geom. Phys. 24, 203-208 (1998)] has shown that the Camassa-Hol m equation is a geodesic flow on the Bott-Virasoro group. In this paper it is shown that the Camassa-Holm equation for the case kappa=0 is the geodesi c spray of the weak Riemannian metric on the diffeomorphism group of the li ne or the circle obtained by right translating the H-1 inner product over t he entire group. This paper uses the right-trivialization technique to rigo rously verify that the Euler-Poincare theory for Lie groups can be applied to diffeomorphism groups. The observation made in this paper has led to phy sically meaningful generalizations of the CH-equation to higher dimensional manifolds. (C) 1999 American Institute of Physics. [S0022-2488(99)02102-7] .