Let us briefly summarize our main results. We have described a generalized version of the powerful tool of normal form theory for Hamiltonian systems. Using this generalized technique, it is now possible to analyze any Hamiltonian that is given as a power series in phase space coordinates. Even if the Hamiltonian is not given in the form of a power series one can always expand into its Taylor series and normalize the truncated expansion. Thus a large variety of Hamiltonian systems can be analyzed in a unified way.
The most important result of a normalization is the derivation of a formal integral of motion that, in general, is different from (and often independent of) the already known integral . That means that one can obtain substantial new information about the system by normalization. Convergence of this formal integral of motion cannot be taken for granted. Addressing this problem, we have suggested some new methods for analyzing the convergence properties of the truncated formal integral. We have found that these quasi-integrals are of physical interest, since their convergence properties reflect many of the characteristics of the corresponding Poincaré plots.
It is certainly necessary to carry the analysis of the convergence of the quasi-integrals further. Many authors [10,33,29] have suggested to study the poles of Padé approximations to the quasi-integrals. The location and the number of poles of these approximants then allow to gauge the properties of .