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Take the planar three-body problem. Or, said a bit differently, take that 'cat' to consist of three point masses moving about in the plane -- a triangle! Fix the center of the mass at the origin by the usual trick. Take G = SO(2). Now take the quotient and you get the cone over the usual Hopf fibration. The points of the sphere in the base spacerepresent oriented similarity classes of triangles. This geometry is at the heart of much modern understanding of the planar three body problem. You can find references in my 2000 paper with Chenciner A remarkable periodic solution of the three-body problem in the case of equal masses' and the geometry explained in some detail in the 1st few pages and in the appendix to my 1996 paper The geometric phase of the three-body problem'. You can download these from http://count.ucsc.edu/~rmont/papers/list.html