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Action-Minimizing Invariant Measures for Tonelli Lagrangians

Action-Minimizing Invariant Measures for Tonelli Lagrangians

Chapter:
(p.18) Chapter Three Action-Minimizing Invariant Measures for Tonelli Lagrangians
Source:
Action-minimizing Methods in Hamiltonian Dynamics (MN-50)
Author(s):
Alfonso Sorrentino
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691164502.003.0003

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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