Livia Corsi, Guido Gentile, Michela Procesi
KAM theory in configuration space
and cancellations in the Lindstedt series
(743K, pdf)
ABSTRACT. The KAM theorem for analytic quasi-integrable anisochronous Hamiltonian
systems yields that the perturbation expansion (Lindstedt series) for
any quasi-periodic solution with Diophantine frequency vector converges.
If one studies the Lindstedt series by following a perturbation
theory approach, 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.
In this paper, 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, and the
interpretation of the underlying symmetries in terms of tree graphs
is much more subtle than in the case of action-angle coordinates.