Weak KAM Theory and Forcing Equivalence
Weak KAM Theory and Forcing Equivalence
This chapter describes weak Kolmogorov-Arnold-Moser (KAM) theory and forcing relation. One change from the standard presentation is that one needs to modify the definition of Tonelli Hamiltonians to allow different periods in the t component. The chapter points out an alternative definition of the alpha function, namely, one can replace the class of minimal measures with the class of closed measures. It then considers a dual setting which corresponds to forward dynamic. It also looks at elementary solutions, static classes, and Peierls barrier. In many parts of the proof, the chapter studies the hyperbolic property of a minimizing orbit, for which the concept of Green bundles is very useful.
Keywords: weak KAM theory, forcing relation, Tonelli Hamiltonians, dual setting, forward dynamic, elementary solutions, static classes, Peierls barrier, Green bundles, Kolmogorov-Arnold-Moser theory
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