KAM theory in configuration space and cancellations in the Lindstedt series

TL;DR

This paper explores the convergence of Lindstedt series in KAM theory by analyzing symmetries and cancellations in Cartesian coordinates, revealing a different mechanism from the traditional action-angle coordinate approach.

Contribution

It provides a novel analysis of the cancellation mechanisms in Lindstedt series within Cartesian coordinates, contrasting with the well-understood symmetries in action-angle coordinates.

Findings

Cancellations enable convergence of Lindstedt series in Cartesian coordinates.

Symmetries responsible for cancellations differ from those in action-angle coordinates.

The interpretation of symmetries via tree graphs is more subtle in Cartesian coordinates.

Abstract

The KAM theorem for analytic quasi-integrable anisochronous Hamiltonian systems yields that the perturbation expansion (Lindstedt series) for quasi-periodic solutions with Diophantine frequency vector converges. If one studies the Lindstedt series, one finds that convergence is ultimately related to the presence of cancellations between contributions of the same perturbation order. In turn, this is due to symmetries in the problem. Such symmetries are easily visualised in action-angle coordinates, where KAM theorem is usually formulated, by exploiting the analogy between Lindstedt series and perturbation expansions in quantum field theory and, in particular, the possibility of expressing the solutions in terms of tree graphs, which are the analogue of Feynman diagrams. If the unperturbed system is isochronous, Moser's modifying terms theorem ensures that an analytic quasi-periodic solution with the same Diophantine frequency vector as the unperturbed Hamiltonian exists for the system obtained by adding a suitable constant (counterterm) to the vector field. Also in this case, one can follow the alternative approach of studying the perturbation expansion for both the solution and the counterterm, and again convergence of the two series is obtained as a consequence of deep cancellations between contributions of the same order. We revisit Moser's theorem, by studying the perturbation expansion one obtains by working in Cartesian coordinates. We investigate the symmetries giving rise to the cancellations which makes possible the convergence of the series. We find that the cancellation mechanism works in a completely different way in Cartesian coordinates. The interpretation of the underlying symmetries in terms of tree graphs is much more subtle than in the case of action-angle coordinates.