The Camassa-Holm equation as a geodesic flow for the $H^1$ right-invariant metric

Details The Camassa-Holm equation as a geodesic flow for the $H^1$ right-invariant metric The Camassa-Holm equation as a geodesic flow for the $H^1$ right-invariant metric

2007-06-26

arxiv.org/abs/0706.3810v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

8 pages, LaTeX, published in ``Topics in Contemporary Differential Geometry, Complex Analysis and Mathematical Physics, Procee

Scientific paper

The fundamental role played by the Lie groups in mechanics, and especially by the dual space of the Lie algebra of the group and the coadjoint action are illustrated through the Camassa-Holm equation (CH). In 1996 Misio{\l}ek observed that CH is a geodesic flow equation on the group of diffeomorphisms, preserving the $H^1$ metric. This example is analogous to the Euler equations in hydrodynamics, which describe geodesic flow for a right-invariant metric on the infinite-dimensional group of diffeomorphisms preserving the volume element of the domain of fluid flow and to the Euler equations of rigid body whith a fixed point, describing geodesics for a left-invariant metric on SO(3). The momentum map and an explicit parametrization of the Virasoro group, related to recently obtained solutions for the CH equation are presented.

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