Geodesic
Viscosity solutions methods for converse KAM theory
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 42 (2008) no. 6, pp. 1047-1064.

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The main objective of this paper is to prove new necessary conditions to the existence of KAM tori. To do so, we develop a set of explicit a-priori estimates for smooth solutions of Hamilton-Jacobi equations, using a combination of methods from viscosity solutions, KAM and Aubry-Mather theories. These estimates are valid in any space dimension, and can be checked numerically to detect gaps between KAM tori and Aubry-Mather sets. We apply these results to detect non-integrable regions in several examples such as a forced pendulum, two coupled penduli, and the double pendulum.

DOI : 10.1051/m2an:2008035
Classification : 37J50, 49L25, 65P10, 70H7
Keywords: Aubry-Mather theory, Hamilton-Jacobi integrability, viscosity solutions 
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Gomes, Diogo A.; Oberman, Adam. Viscosity solutions methods for converse KAM theory. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 42 (2008) no. 6, pp. 1047-1064. doi : 10.1051/m2an:2008035. http://geodesic.mathdoc.fr/articles/10.1051/m2an:2008035/

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