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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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Forcing equivalence between kissing cylinders

Forcing equivalence between kissing cylinders

(p.133) Chapter Twelve Forcing equivalence between kissing cylinders
Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom

Kaloshin Vadim

Zhang Ke

Princeton University Press

This chapter formulates and proves the jump mechanism. It constructs a variational problem which proves forcing equivalence for the original Hamiltonian using Definition 6.18. It first constructs a special variational problem for the slow mechanical system. A solution of this variational problem is an orbit “jumping” from one homology class to the other. The chapter then modifies this variational problem for the fast time-periodic perturbation of the slow mechanical system. This is achieved by applying the perturbative results established in Chapter 7. Recall the original Hamiltonian system near a double resonance can be brought to a normal form and this normal form, in turn, is related to the perturbed slow system through coordinate change and energy reduction. The variational problem for the perturbed slow system can then be converted to a variational problem for the original.

Keywords:   jump mechanism, variational problem, forcing equivalence, Hamiltonian system, slow mechanical system, homology class, perturbed slow system

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