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Action-minimizing Methods in Hamiltonian Dynamics (MN-50)An Introduction to Aubry-Mather Theory$
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Alfonso Sorrentino

Print publication date: 2015

Print ISBN-13: 9780691164502

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691164502.001.0001

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Action-Minimizing Curves for Tonelli Lagrangians

Action-Minimizing Curves for Tonelli Lagrangians

Chapter:
(p.48) Chapter Four Action-Minimizing Curves for Tonelli Lagrangians
Source:
Action-minimizing Methods in Hamiltonian Dynamics (MN-50)
Author(s):

Alfonso Sorrentino

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691164502.003.0004

This chapter discusses the notion of action-minimizing orbits. In particular, it defines the other two families of invariant sets, the so-called Aubry and Mañé sets. It explains their main dynamical and symplectic properties, comparing them with the results obtained in the preceding chapter for the Mather sets. The relation between these new invariant sets and the Mather sets is described. As a by-product, the chapter introduces the Mañé's potential, Peierls' barrier, and Mañé's critical value. It discusses their properties thoroughly. In particular, it highlights how this critical value is related to the minimal average action and describes these new concepts in the case of the simple pendulum.

Keywords:   action-minimizing orbits, invariant sets, Aubry set, Mañé set, Maher sets, Mañé's potential, Peierls' barrier, Mañé's critical value

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