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Books on the topic 'KAM and Nekhoroshev theory'

Author: Grafiati

Published: 4 June 2021

Last updated: 31 January 2023

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1

Giancarlo, Benettin, Henrard J, Kuksin Sergej B. 1955-, Giorgilli Antonio, Centro internazionale matematico estivo, and European Mathematical Society, eds. Hamiltonian dynamics theory and applications: Lectures given at the C.I.M.E.-E.M.S. Summer School, held in Cetraro, Italy, July 1-10, 1999. Berlin: Springer, 2005.

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2

Kappeler, Thomas. KdV & KAM. Berlin: Springer, 2003.

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3

Jürgen, Pöschel, ed. KdV & KAM. Berlin: Springer, 2003.

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4

Lazutkin, Vladimir F. KAM theory andsemiclassical approximations to eigenfunctions. Berlin: Springer-Verlag, 1993.

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5

Luo, Albert C. J., and Valentin Afraimovich, eds. Hamiltonian Chaos Beyond the KAM Theory. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-12718-2.

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6

Lazutkin, V. F. KAM theory and semiclassical approximations to eigenfunctions. Berlin: Springer-Verlag, 1993.

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7

Lazutkin, Vladimir F. KAM Theory and Semiclassical Approximations to Eigenfunctions. Berlin, Heidelberg: Springer Berlin Heidelberg, 1993.

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8

Lazutkin, Vladimir F. KAM Theory and Semiclassical Approximations to Eigenfunctions. Berlin, Heidelberg: Springer Berlin Heidelberg, 1993. http://dx.doi.org/10.1007/978-3-642-76247-5.

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9

González-Enríquez, A. Singularity theory for non-twist KAM tori. Providence, Rhode Island: American Mathematical Society, 2013.

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10

Valentin, Afraimovich, and SpringerLink (Online service), eds. Hamiltonian Chaos Beyond the KAM Theory: Dedicated to George M. Zaslavsky (1935–2008). Berlin, Heidelberg: Springer-Verlag Berlin Heidelberg, 2011.

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11

Boccaletti, D. Theory of orbits. Berlin: Springer-Verlag, 1996.

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12

G, Pucacco, ed. Theory of orbits. 3rd ed. Berlin: Springer, 2004.

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13

Boccaletti, D. Theory of orbits. 2nd ed. Berlin: Springer, 2001.

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14

Benettin, Giancarlo, Jacques Henrard, Sergej B. Kuksin, and Antonio Giorgilli. Hamiltonian Dynamics - Theory and Applications: Lectures Given at the C. I. M. E. Summer School Held in Cetraro, Italy, July 1-10 1999. Springer London, Limited, 2010.

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15

Henrard, Jacques, Giancarlo Benettin, and Sergei Kuksin. Hamiltonian Dynamics - Theory and Applications: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 1-10, 1999 (Lecture Notes in Mathematics / Fondazione C.I.M.E., Firenze). Springer, 2005.

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16

Sorrentino, Alfonso. From KAM Theory to Aubry-Mather Theory. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691164502.003.0002.

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This chapter discusses an illustrative example, namely the properties of invariant probability measures and orbits on KAM tori (or more generally, on invariant Lagrangian graphs). This will prepare the ground for understanding the main ideas and techniques that will be developed in the following chapters, without several technicalities that might be confusing to a neophyte.

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17

Pöschel, Jürgen, and Thomas Kappeler. KdV and KAM. Springer London, Limited, 2013.

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18

Kam, V. Kam Solutions Manual to Accompany Accounting Theory. John Wiley & Sons Inc, 1986.

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19

Sorrentino, Alfonso. The Hamilton-Jacobi Equation and Weak KAM Theory. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691164502.003.0005.

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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.

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20

Afraimovich, Valentin, and Albert C. J. Luo. Hamiltonian Chaos Beyond the KAM Theory: Dedicated to George M. Zaslavsky. Springer, 2014.

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21

Kappeler, Thomas, and Jürgen Pöschel. KdV & KAM (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics). Springer, 2003.

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22

Dumas, H. S. The KAM Story: A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov-Arnold-Moser Theory. World Scientific Publishing Company, 2014.

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23

Boccaletti, Dino, and Giuseppe Pucacco. Theory of Orbits: Volume 2: Perturbative and Geometrical Methods (Astronomy and Astrophysics Library). Springer, 2004.

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24

Pucacco, G., and D. Boccaletti. Theory of Orbits: Volume 1: Integrable Systems and Non-perturbative Methods (Astronomy and Astrophysics Library). Springer, 2003.

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25

Nolte, David D. From Butterflies to Hurricanes. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805847.003.0009.

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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.

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26

Sorrentino, Alfonso. Action-minimizing Methods in Hamiltonian Dynamics (MN-50). Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691164502.001.0001.

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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.

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27

Mann, Peter. The Structure of Phase Space. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0023.

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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.

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28

Kaloshin, Vadim, and Ke Zhang. Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom. Princeton University Press, 2020. http://dx.doi.org/10.23943/princeton/9780691202525.001.0001.

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Arnold diffusion, which concerns the appearance of chaos in classical mechanics, is one of the most important problems in the fields of dynamical systems and mathematical physics. Since it was discovered by Vladimir Arnold in 1963, it has attracted the efforts of some of the most prominent researchers in mathematics. The question is whether a typical perturbation of a particular system will result in chaotic or unstable dynamical phenomena. This book provides the first complete proof of Arnold diffusion, demonstrating that that there is topological instability for typical perturbations of five-dimensional integrable systems (two and a half degrees of freedom). This proof realizes a plan John Mather announced in 2003 but was unable to complete before his death. The book follows Mather's strategy but emphasizes a more Hamiltonian approach, tying together normal forms theory, hyperbolic theory, Mather theory, and weak KAM theory. Offering a complete, clean, and modern explanation of the steps involved in the proof, and a clear account of background material, the book is designed to be accessible to students as well as researchers. The result is a critical contribution to mathematical physics and dynamical systems, especially Hamiltonian systems.

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