TITLE:
KAM Theory Without Action-Angle Coordinates
AUTHORS:
Alejandra Gonzalez(1), Angel Jorba(1), Rafael de la Llave(2)
and Jordi Villanueva(3)
(1) Departament de Matematica Aplicada i Analisi,
Universitat de Barcelona,
Gran Via 585, 08007 Barcelona (Spain)
E-mails: gonzalez@maia.ub.es, angel@maia.ub.es
(2) Department of Mathematics,
University of Texas at Austin,
Austin, TX 78712 (USA).
E-mail: llave@math.utexas.edu
(3) Dept. de Matematica Aplicada I (ETSEIB),
Universitat Politecnica de Catalunya,
Diagonal 647, 08028 Barcelona (Spain).
E-mail: jordi@tere.upc.es
ABSTRACT:
The classical KAM methods, strongly supported on the use of canonical
transformations in the action-angle context, are not efficient to be
applied to a wide range of systems in which the Hamiltonian is known
(for instance) written in Cartesian coordinates.
In this communication we present some ideas to deal with KAM theory
using ``parameterizations'' instead of ``transformations'' and
``graphs'', which we think is an efficient way to work with a more
general class of Hamiltonian systems than the classical methods (in
particular, for systems motivated by real world problems). With the
present approach, we can extend several well-known results of KAM
theory to these systems, even when the classical statements are
difficult to be applied.