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Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom(AMS-208)$
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Vadim Kaloshin and Ke Zhang

Print publication date: 2020

Print ISBN-13: 9780691202525

Published to Princeton Scholarship Online: May 2021

DOI: 10.23943/princeton/9780691202525.001.0001

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Perturbative weak KAM theory

Perturbative weak KAM theory

(p.66) Chapter Seven Perturbative weak KAM theory
Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom

Kaloshin Vadim

Zhang Ke

Princeton University Press

This chapter explores perturbation aspects of the weak Kolmogorov-Arnold-Moser (KAM) theory. By perturbative weak KAM theory, we mean two things. How do the weak KAM solutions and the Mather, Aubry, and Mañé sets respond to limits of the Hamiltonian? How do the weak KAM solutions change when we perturb a system, in particular, what happens when we perturb (1) completely integrable systems, and (2) autonomous systems by a time-periodic perturbation? The chapter states and proves results in both aspects, as a technical tool for proving forcing equivalence. It derives a special Lipshitz estimate of weak KAM solutions for perturbations of autonomous systems. The proof relies on semi-concavity of weak KAM solution.

Keywords:   time-periodic perturbation, weak KAM theory, Kolmogorov-Arnold-Moser theory, perturbative weak KAM theory, weak KAM solutions, Hamiltonian, forcing equivalence, Lipshitz estimate, autonomous systems, integrable systems

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