The Transition to Chaos
In Conservative Classical Systems: Quantum Manifestations
Published on 17. April 2013
XVI, 551 pages
978-1-4757-4352-4 (ISBN)
Description
resonances. Nonlinear resonances cause divergences in conventional perturbation expansions. This occurs because nonlinear resonances cause a topological change locally in the structure of the phase space and simple perturbation theory is not adequate to deal with such topological changes. In Sect. (2.3), we introduce the concept of integrability. A sys tem is integrable if it has as many global constants of the motion as degrees of freedom. The connection between global symmetries and global constants of motion was first proven for dynamical systems by Noether [Noether 1918]. We will give a simple derivation of Noether's theorem in Sect. (2.3). As we shall see in more detail in Chapter 5, are whole classes of systems which are now known to be inte there grable due to methods developed for soliton physics. In Sect. (2.3), we illustrate these methods for the simple three-body Toda lattice. It is usually impossible to tell if a system is integrable or not just by looking at the equations of motion. The Poincare surface of section provides a very useful numerical tool for testing for integrability and will be used throughout the remainder of this book. We will illustrate the use of the Poincare surface of section for classic model of Henon and Heiles [Henon and Heiles 1964].
Reviews / Votes
From the reviews of the second edition:
"This book is an expanded and updated version . from a previous edition and reviews results on the manifestation of chaos in classical and quantum mechanics. . A very wide range of topics is covered in the book, which thus can be used as preliminary reading for research areas . . The book can also be considered as a helpful companion both for mathematicians and for physicists. . Many technical details and background notions can be found in a rich complement of appendices." (Guido Gentile, Mathematical Reviews, Issue 2006 c)
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Content
1. Overview.- I Classical Systems.- 2. Fundamental Concepts.- 3. Area Preserving Maps.- 4. Global Properties.- II Quantum Systems.- 5. Quantum Integrability.- 6. Random Matrix Theory.- 7. Observed Spectra.- 8. Semi-Classical Theory.- 9. Driven Systems.- III Stochastic Systems.- 10. Stochastic Systems.- IV Appendices.- A. Classical Mechanics.- A.1 Newton's Equations.- A.2 Lagrange's Equations.- A.3 Hamilton's Equations.- A.4 The Poisson Bracket.- A.5 Phase Space Volume Conservation.- A.6 Action-Angle Variables.- A.7 Hamilton's Principle Function.- A.8 References.- B. Simple Models.- B.1 The Pendulum.- B.2 Double Well Potential.- B.3 Infinite Square Well Potential.- B.4 One-Dimensional Hydrogen.- C.Renormalization Integral.- C.3 References.- D.The Moyal Bracket.- D.1 The Wigner Function.- D.2 Ordering of Operators.- D.3 Moyal Bracket.- D.4 References.- E. SU(3).- E.1 Special Unitary Groups.- E.2 References.- F.Space-Time Symmetries.- F.1 Linear and Antilinear Operators.- F.2 Infinitesimal Transformations.- F.3 Discrete Transformations.- F.4 References.- G. GOE Spectral Statistics.- G.5 References.- H. COE Spectral Statistics.- H.4 References.- I. Lloyd's Model.- L.1 Localization Length.- L.2 References.- J. Hydrogen in Parabolic Coordinates.- J.1 The Schrodinger Equation.- J.2 One-dimensional Hydrogen.- J.3 References.- Author Index.
