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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 Invariant Measures for Tonelli Lagrangians

Action-Minimizing Invariant Measures for Tonelli Lagrangians

(p.18) Chapter Three Action-Minimizing Invariant Measures for Tonelli Lagrangians
Action-minimizing Methods in Hamiltonian Dynamics (MN-50)

Alfonso Sorrentino

Princeton University Press

This chapter discusses the notion of action-minimizing measures, recalling the needed measure–theoretical material. In particular, this allows the definition of a first family of invariant sets, the so-called Mather sets. It discusses their main dynamical and symplectic properties, and introduces the minimal average actions, sometimes called Mather's α‎- and β‎-functions. A thorough discussion of their properties (differentiability, strict convexity or lack thereof) is provided and related to the dynamical and structural properties of the Mather sets. The chapter also describes these concepts in a concrete physical example: the simple pendulum.

Keywords:   action-minimizing measure, Maher sets, invariant sets, differentiability, strict convexity, pendulum

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