6.5  Summary

The goal of perturbation theory is to relate aspects of the motions of a given system to those of a nearby solvable system. Perturbation theory can be used to predict features such as the size and location of the resonance islands and chaotic zones.

With perturbation analysis we obtain an approximation to the evolution of a system by relating the evolution of the system to that of a different system that, when approximated, can be exactly solved. We can carry this exact solution of the approximate problem back to the original system to obtain an approximate solution of our original problem. The strategy of canonical perturbation theory is to make canonical transformations that eliminate terms in the Hamiltonian that impede solution. Formulation of perturbation theory in terms of Lie series is especially convenient.

We can use first-order perturbation theory to analyze the motion of the undriven pendulum as a free rotor to which gravity is added. In this analysis we find that a small denominator in the series limits the range of applicability of the perturbative solution to regions that are away from the resonant oscillation region.

In higher-order perturbation theory for the pendulum we discover the problem of secular terms, terms that produce error that grow with time. The appearance of secular terms can be avoided by keeping track of how the frequencies change as perturbations are included. In canonical perturbation theory secular terms can be avoided by associating the average part of the perturbation with the solvable part of the Hamiltonian.

In carrying out canonical perturbation theory in higher dimensions we find that the problem of small denominators is more serious. Small denominators arise near every commensurability, and commensurabilities are common. Small denominators can be locally avoided near particular commensurabilities by incorporating the offending terms into the solvable part of the Hamiltonian. If the resonances are isolated, the resulting resonance Hamiltonian is still solvable. In many cases the resonance Hamiltonian is well approximated by a pendulum-like Hamiltonian. A global picture can be constructed by stitching together the solutions for each resonance region constructed separately.

If two resonance regions overlap -- that is, if the sum of the half-widths of the resonance regions exceeds their separation -- then large-scale chaos ensues. The chaotic regions associated with the separatrices of the overlapping resonances become connected. When the resonances are well approximated by pendulum-like resonances a simple analytic criterion for the appearance of large-scale chaos can be developed.

Higher-order perturbative descriptions can be developed to describe islands that do not correspond to particular terms in the Hamiltonian, secondary resonances, bifurcations, and so on. The theory can be extended to describe as much detail as one wishes.