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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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The Hamilton-Jacobi Equation and Weak KAM Theory

The Hamilton-Jacobi Equation and Weak KAM Theory

Chapter:
(p.76) Chapter Five The Hamilton-Jacobi Equation and Weak KAM Theory
Source:
Action-minimizing Methods in Hamiltonian Dynamics (MN-50)
Author(s):

Alfonso Sorrentino

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691164502.003.0005

This chapter describes another interesting approach to the study of invariant sets provided by the so-called weak KAM theory, developed by Albert Fathi. This approach can be considered as the functional analytic counterpart of the variational methods discussed in the previous chapters. The starting point is the relation between KAM tori (or more generally, invariant Lagrangian graphs) and classical solutions and subsolutions of the Hamilton–Jacobi equation. It introduces the notion of weak (non-classical) solutions of the Hamilton–Jacobi equation and a special class of subsolutions (critical subsolutions). In particular, it highlights their relation to Aubry–Mather theory.

Keywords:   invariant sets, KAM theory, Albert Fathi, MAK tori, Hamilton–Jacobi equation, Aubry–Mather theory

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