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I posted this on stack exchange but had no joy, perhaps someone here can answer : The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group. For the famous $KdV$ equations these equations are given on the Virasoro-Bott group, a central extension of the group of diffeomorphisms of the circle. Since then other equations of fluid mechanics have been constructed on this group (with a different metric).

it is easy to understand, intuitively, why an equation of fluid dynamics would be a geodesic equation on a group of diffeomorphisms, but I don't see where the central extension comes in. How was this discovered? What was the motivation for using the central extension of such a lie group to describe these equations?

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    $\begingroup$ Not a real answer, but have you read the book by Khesin and Wendt: The Geometry of Infinite-Dimensional Groups. In there a lot of references and a rough sketch of the constructions are given. I figure that one may find some clues along the lines you are interested in in the material there. $\endgroup$
    – Alexander Schmeding
    Commented Apr 24, 2016 at 20:22
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    $\begingroup$ I have actually taken a look at that book and although the construction there is very clear there is no real motivation given. It is possible I am looking for meaning where there isn't any. Adding in the central extension is what gives the dispersion term so I guess it could have just been "noticed", it just weirds me out that such a well studied equation would simply pop out on a unique central extension with an L^2 metric for no particular reason... Anyway, thank you for your help! $\endgroup$
    – R Mary
    Commented Apr 25, 2016 at 12:22
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    $\begingroup$ Could it somehow be related to the central extension involving the Schwarzian through a cocyle and the connection between the Schwarzian and the velocity of soliton solutions of the KdV equation, a velocity which is related to deviation of the solns from Moebius transformations? mathoverflow.net/questions/38105/… and mathoverflow.net/questions/145555/… $\endgroup$
    – Tom Copeland
    Commented Apr 26, 2016 at 4:07

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