Phase space concepts

By 1892, Henri Poincare already had published his landmark work in which he advanced the theses that differential equations should be viewed as geometric objects, in particular, as vector fields on manifolds, and that questions concerning the long-term stability of a dynamical system might be attacked by studying the topological properties of these objects. His work even led him to recognize the extraordinarily complicated behavior of orbits in the vicinity of a separatrix, what today they would call chaos. In these lectures the author looks into this geometric approach to the study of Hamiltonian dynamical systems, especially in connection with the kinds of problems which arises in accelerator orbit theory. The modest goals here will be to delineate the idea of invariant tori in phase space, to define and illustrate the importance of resonant orbits and their separatrices as structures for organizing dynamics, and to touch upon the meaning of chaotic orbits in nonintegrable systems. Further, the author hopes to lessen the impression which some may have acquired during their formal education that classical mechanics is a dull, closed subject with no mysteries left to explore. 26 references, 15 figures