3 - Conservative dynamical systems

Summary

Integrable conservative systems: symmetry, invariance, conservation laws, and motion on invariant tori in phase space

Systems that show instability with respect to small changes in initial data were neither studied nor taught in most physics departments until nearly two decades after Lorenz's discovery of the butterfly effect. The reason for this is in part that classical mechanics was regarded by many physicists as a dead subject and was often taught mainly as preparation for the study of quantum mechanics: dissipative systems were largely ignored and the study of Hamiltonian mechanics was restricted to a very special case, that of completely integrable (or just ‘integrable’) systems. Mechanics was typically taught as if there were no systems other than the integrable ones. One reason for this was the very strong influence of progress that had been made in quantum theory by studying systems that were completely solvable by using symmetry. Whenever you can solve a dynamics problem by symmetry, then the system is called integrable, or ‘completely integrable’. This terminology has been used in quantum as well as in classical mechanics, and for the same reason: in both cases, the method proceeds by finding a (finite) complete set of commuting infinitesimal generators (complete set of commuting operators).