Integrability

The problem of integrability in classical mechanics has been a seminal one. Motivated by celestial mechanics, it has stimulated a wealth of analytical methods and results. For example, as we have discussed in Chapter 2, the weaker requirement of only approximate integrability over finite times, or the existence of integrable regions in the phase space of a globally nonintegrable system, has led to the development of classical perturbation theory, with all its important achievements. However, deciding whether a given Hamiltonian system is globally integrable still remains a difficult task, for which a general constructive framework is lacking.