*Alfonso Sorrentino*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164502
- eISBN:
- 9781400866618
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164502.003.0005
- Subject:
- Mathematics, Applied Mathematics

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 ...
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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.Less

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.

*Alfonso Sorrentino*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164502
- eISBN:
- 9781400866618
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164502.001.0001
- Subject:
- Mathematics, Applied Mathematics

John Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical ...
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John Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical mechanics. His novel approach—known as Aubry–Mather theory—singles out the existence of special orbits and invariant measures of the system, which possess a very rich dynamical and geometric structure. In particular, the associated invariant sets play a leading role in determining the global dynamics of the system. This book provides a comprehensive introduction to Mather's theory, and can serve as an interdisciplinary bridge for researchers and students from different fields seeking to acquaint themselves with the topic. Starting with the mathematical background from which Mather's theory was born, the book first focuses on the core questions the theory aims to answer—notably the destiny of broken invariant KAM tori and the onset of chaos—and describes how it can be viewed as a natural counterpart of KAM theory. The book achieves this by guiding readers through a detailed illustrative example, which also provides the basis for introducing the main ideas and concepts of the general theory. It then describes the whole theory and its subsequent developments and applications in their full generality.Less

John Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical mechanics. His novel approach—known as Aubry–Mather theory—singles out the existence of special orbits and invariant measures of the system, which possess a very rich dynamical and geometric structure. In particular, the associated invariant sets play a leading role in determining the global dynamics of the system. This book provides a comprehensive introduction to Mather's theory, and can serve as an interdisciplinary bridge for researchers and students from different fields seeking to acquaint themselves with the topic. Starting with the mathematical background from which Mather's theory was born, the book first focuses on the core questions the theory aims to answer—notably the destiny of broken invariant KAM tori and the onset of chaos—and describes how it can be viewed as a natural counterpart of KAM theory. The book achieves this by guiding readers through a detailed illustrative example, which also provides the basis for introducing the main ideas and concepts of the general theory. It then describes the whole theory and its subsequent developments and applications in their full generality.

*David D. Nolte*

- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198805847
- eISBN:
- 9780191843808
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198805847.003.0009
- Subject:
- Physics, History of Physics

Half a century after Poincaré first glimpsed chaos in the three-body problem, the great Russian mathematician Andrey Kolmogorov presented a sketch of a theorem that could prove that orbits are ...
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Half a century after Poincaré first glimpsed chaos in the three-body problem, the great Russian mathematician Andrey Kolmogorov presented a sketch of a theorem that could prove that orbits are stable. In the hands of Vladimir Arnold and Jürgen Moser, this became the Kolmo–Arnol–Mos (KAM) theory of Hamiltonian chaos. This chapter shows how KAM theory fed into topology in the hands of Stephen Smale and helped launch the new field of chaos theory. Edward Lorenz discovered chaos in numerical models of atmospheric weather and discovered the eponymous strange attractor. Mathematical aspects of chaos were further developed by Mitchell Feigenbaum studying bifurcations in the logistic map that describes population dynamics.Less

Half a century after Poincaré first glimpsed chaos in the three-body problem, the great Russian mathematician Andrey Kolmogorov presented a sketch of a theorem that could prove that orbits are stable. In the hands of Vladimir Arnold and Jürgen Moser, this became the Kolmo–Arnol–Mos (KAM) theory of Hamiltonian chaos. This chapter shows how KAM theory fed into topology in the hands of Stephen Smale and helped launch the new field of chaos theory. Edward Lorenz discovered chaos in numerical models of atmospheric weather and discovered the eponymous strange attractor. Mathematical aspects of chaos were further developed by Mitchell Feigenbaum studying bifurcations in the logistic map that describes population dynamics.

*Peter Mann*

- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198822370
- eISBN:
- 9780191861253
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198822370.003.0023
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter introduces the reader to canonical perturbation theory as a tool for studying near-integrable systems. Many problems in physics and chemistry do not have exact analytical solutions; ...
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This chapter introduces the reader to canonical perturbation theory as a tool for studying near-integrable systems. Many problems in physics and chemistry do not have exact analytical solutions; these systems are in direct opposition to integrable systems and action-angle variables. The chapter starts by considering tiny perturbations to integrable Hamiltonians. Poincaré in 1893 claimed this was the fundamental question of classical mechanics and, fittingly, Hamilton–Jacobi theory is the starting point. The chapter develops Poincaré’s fundamental equation as well as Delaunay’s small divisor problem. Resonant, near–resonant and non-resonant tori are investigated in the context of Poincaré’s theorem and KAM theory is described in detail. Chaos and Poincaré maps are presented before discussing determinism, deterministic chaos and Laplace’s demon.Less

This chapter introduces the reader to canonical perturbation theory as a tool for studying near-integrable systems. Many problems in physics and chemistry do not have exact analytical solutions; these systems are in direct opposition to integrable systems and action-angle variables. The chapter starts by considering tiny perturbations to integrable Hamiltonians. Poincaré in 1893 claimed this was the fundamental question of classical mechanics and, fittingly, Hamilton–Jacobi theory is the starting point. The chapter develops Poincaré’s fundamental equation as well as Delaunay’s small divisor problem. Resonant, near–resonant and non-resonant tori are investigated in the context of Poincaré’s theorem and KAM theory is described in detail. Chaos and Poincaré maps are presented before discussing determinism, deterministic chaos and Laplace’s demon.