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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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Generic properties of mechanical systems on the two-torus

Generic properties of mechanical systems on the two-torus

Chapter:
(p.146) (p.147) Chapter Thirteen Generic properties of mechanical systems on the two-torus
Source:
Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom
Author(s):

Kaloshin Vadim

Zhang Ke

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691202525.003.0013

This chapter focuses on proving generic properties of the minimizing orbits of the slow mechanical system. It first proves Theorem 4.5 concerning non-critical but bounded energy, before proving Proposition 4.6 concerning the very high energy. The chapter then proves Proposition 4.7 concerning the critical energy. The proof of Theorem 4.5 consists of three steps. The first proves a Kupka-Smale-like theorem about non-degeneracy of periodic orbits. The second shows that a non-degenerate locally minimal orbit is always hyperbolic. The third finishes the proof by proving the finite local families obtained from the second step are “in general position,” and therefore there are at most two global minimizers for each energy.

Keywords:   slow mechanical system, critical energy, non-critical energy, Kupka-Smale theorem, periodic orbits, local families, global minimizers

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