METHODS FOR THE ANALYSIS OF THE LINDSTEDT SERIES FOR KAM TORI AND RENORMALIZABILITY IN CLASSICAL MECHANICS: A review with Some Applications

Abstract

This paper consists in a unified exposition of methods and techniques of the renormalization group approach to quantum field theory applied to classical mechanics, and in a review of results: (1) a proof of the KAM theorem, by studing the perturbative expansion (Lindstedt series) for the formal solution of the equations of motion; (2) a proof of a conjecture by Gallavotti about the renormalizability of isochronous hamiltonians, i.e. the possibility to add a term depending only on the actions in a hamiltonian function not verifying the anisochrony condition so that the resulting hamiltonian is integrable. Such results were obtained first by Eliasson; however the difficulties arising in the study of the perturbative series are very similar to the problems which one has to deal with in quantum field theory, so that the use the methods which have been envisaged and developed in the last twenty years exactly in order to solve them allows us to obtain unified proofs, both conceptually and technically. In the final part of the review, the original work of Eliasson is analyzed and exposed in detail; its connection with other proofs of the KAM theorem based on his method is elucidated.

Aspects of Ergodic, Qualitative and Statistical Theory of Motion

“…More recently a new proof of the theorem has been derived by Eliasson,[El96]: we have followed the interpretation given to it in [Ga94]. The proof presented in this section deals with a special case but it can be done in full generality along the same lines, see [GM96] and Section §8.4. The technique used is inspired to the perturbation theory in quantum field theory and renormalization group, [Ga01].…”

“…Fix p ≤ 0 and let G = [a, b] be an interval verifying what we shall call §8.4: Appendix A8.4: Weakening the strong Diophantine condition the shifting of the exiting branch of the self-energy graphs (as originally remarked in [CF94]; see [GM96] for an implementation within a formalism closer to the one described here). The generalization to perturbations depending also on the action variables requires some minor extensions of the cancellations described in Section §8.1, which can be found in [CF96] and in [GM96]; also the case of unperturbed Hamiltonians K(A) with a more general dependence on the action variables can be dealt with provided that one has det ∂ 2 A K(A) = 0 (anisochrony condition). But no real new difficulty arises; we shall not discuss further such a case here, for which we refer to the original papers.…”

Quasiperiodic motions in dynamical systems: Review of a renormalization group approach

“…Note that in the case (3.2) the unperturbed solution is u 0 = (α 0 + ω(A 0 )t, A 0 ), with ω(A) = ∂ A H 0 (A), hence it is still of the form c 0 + ωt with n = 2p and ω i = 0 for i ≥ n + 1. However, strictly speaking the Hamilton equation are not of the form (1.1), so the analysis should be suitably adapted -see [47].…”

“…The method is widely inspired to the original work of Eliasson [29] and, even more, to its reinterpretation given by Gallavotti [32]. The deep analogy with quantum field theory was stressed and used to full extend in subsequent papers; see for instance [46,47].…”

Non-commutative Renormalization

“…For instance they understood secular averages in celestial mechanics as a kind of renormalization [16]- [17]. In classical mechanics, small denominators play the role of high frequencies or ultraviolet divergences in ordinary RG.…”