Steve Smale and Geometric Mechanics

Thus, one can say-perhaps with only a slight danger of oversimplification- that reduction theory synthesises the work of Smale, Arnold (and their predecesors of course) into a bundle, with Smale as the base and Arnold as the fiber. This bundle has interesting topology and carries mechanical connections (with associated Chern classes and Hannay-Berry phases) and has interesting singularities (Arms, Marsden, and Moncrief, Guillemin and Sternberg, Atiyab, and otbers). We will describe some of these features later.