Steve Smale and Geometric Mechanics

Marsden, J. E.

In Proceedings of the Smalefest, Springer-Verlag, (M. Hirsch, J. E. Marsden and M. Shub, eds.), (1993) 499–516


One can say—perhaps with only a slight danger of oversimplification—that reduction theory synthesises the work of Smale, Arnold (and their predecessors 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, Atiyah, and others). We will describe some of these features later.