Abstract
Misiolek [J. Geom. Phys. 24, 203–208 (1998)] has shown that the Camassa–Holm equation is a geodesic flow on the Bott–Virasoro group. In this paper it is shown that the Camassa–Holm equation for the case κ=0 is the geodesic spray of the weak Riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the H1 inner product over the entire group. This paper uses the right-trivialization technique to rigorously verify that the Euler–Poincaré theory for Lie groups can be applied to diffeomorphism groups. The observation made in this paper has led to physically meaningful generalizations of the CH-equation to higher dimensional manifolds.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
