Interesting Books related to the subject matter for
Nonlinear Mechanics
Spring, 2004

• Introduction to Dynamics, by Percival and Richards (Cambridge Univ. Press)
  This is a small, paperback book that acts as an introduction to Hamiltonian dynamics for advanced undergraduates in the (elite branch of the) British system. It begins in a very elementary way, and advances to some reasonably profound material. It never tries to be more general than necessary to get the point across, and almost everything is restricted to as few dimensions as possible, often only 1, but sometimes 2. It is inexpensive and definitely worth having.
• Classical Dynamics, by José and Saletan (Cambridge Univ. Press)
  This has been the text for the first, graduate course in classical mechanics, Phys. 503, as taught by the department the last several years. That course certainly covers only some relatively small portion of it. However, it is a very good text, and does the best job of presenting the geometrical background which we will use extensively during the course.
  You may well already own this book; if not, then you should.
• Regular and chaotic dynamics, 2nd Edn., by Lichtenberg and Lieberman (Springer-Verlag)
  This is a rather more detailed book, almost a monograph rather than a textbook, on the subject of nonlinear and chaotic dynamics, with lots of mathematical detail, proofs, and assumptions about what you already know. However, it has many good features, and we hope to talk about some of the interesting problems it discusses.

• Deterministic Chaos, by Schuster (VCH, 1995)
  A good book at much the same sort of level as the texts above, although surely somewhat lower than L&L. On the other hand, it spends almost all of its time on non-Hamiltonian systems that exhibit chaotic motion, including much time on discrete ones. Has marvellous pictures.

• Nonlinear Physics with Maple for Scientists and Engineers, by Enns and McGuire (Birkhäuser, 2000 or 1997)
  A very nice book, with lots of "canned" Maple programs for numerical investigations of the things that it discusses [on CD-ROM]. The coverage is rather good, but the level is definitely below that of the Hamilton-Jacobi theory; rather it wants to explain facts rather than theories. Either edition is OK for our purposes, although the computer code is somewhat updated and expanded in the later one.

• Indra's Pearls, The Vision of Felix Klein, by Mumford, Series, and Wright (Cambridge, 2002)
  An unusual book, which assumes almost no pre-requisites at all, begins "well before" what I would consider the beginning, but nonetheless pushes the very curious student along so that a quite marvellous understanding of fractals, computer simulations, transformation theory in the complex plane, and other related things is had by the time one works entirely through the book.

• Introduction to Mechanics and Symmetry, by Marsden and Ratiu (Springer-Verlag, 1994)
  This is quite an interesting and useful book; however, do not be fooled by the title. It is, perhaps, an introduction to the next book on the list below, but surely not to anything less grandiose. Its mathematical level is very high, and thorough. It is definitely a good place to go to begin to get more mathematically-rigorous understandings of the symplectic structures which actually underlie Hamiltonian dynamics and canonical transformations.

• Foundations of Mechanics, by Abraham and Marsden [2nd Edn.] (Benjamin, ???)
  This is a thoroughly rigorous, mathematical explication of the foundations of Hamiltonian mechanics. It is a difficult "slog" through it. The second edition is very much better than the first one.

• Mathematical Methods of Classical Mechanics, by Arnol'd (Springer-Verlag, 1980)
  This is a book more dedicated to pushing forward to new heights, rather than filling in the foundational depths, as did the previously-listed book. Its mathematical level is at least as high, and the required background reading is large; nonetheless, it is definitely a text that any researcher in this area must have read.

• Ergodic Problems of Classical Mechanics, V.I. Arnol'd and A. Avez (W.A. Benjamin, 1968)
  This is the reference from which everyone takes their comments concerning the structure of the phase space for non-integrable systems in the neighborhood of becoming chaotic.

• Advanced Mathematical Methods with Maple, by Richards (Cambridge, 2002)
  Covers much more ground than just nonlinear dynamics, and gives much less explicit direction concerning the use of Maple than the one above; nonetheless, it has some excellent later chapters on nonlinear dynamics, with good discussions of the techniques and understandings needed, although certainly abjuring proofs. It also provides an English translation of Poincaré's article, "On the Stability of the Solar System," originally published in Nature, in June, 1898.

• Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, by Guckenheimer and Holmes (Springer, 1983)
  I am not too familiar with it, but its level of mathematical rigor is at least as high as our text, by Lichtenberg and Lieberman.

• Solitons: an introduction, by Drazin and Johnson (Cambridge, 1989)
  A rather straight-forward, easy-reading text with many examples and details of the behavior of solutions of nonlinear, dissipative pde's that allow soliton solutions, especially for the KdV and sine-Gordon equations. Also considerable, introductory-level material on the Gel'fand-Marchenko inverse scattering equations.

• Theory of Orbits, Vols. 1 and 2, by D. Boccaletti and G. Pucacco (Springer 1996)
  Is an excellent, reasonably modern compendium of many of the facts about modern approaches to celestial mechanics. The first volume considers integrable systems and non-perturbative methods, while the second one deals with perturbation theories, homoclinic tangles, etc. It gives many derivations, but is more into being a compendium of information and references.

• Introduction to Applied Nonlinear Dynamical Systems and Chaos, by Stephen Wiggins (Springer 1990 [corrections in 1996])
  This quite mathematical text seems the best I have so far found on the problems involved with the homoclinic tangle, the Smale horseshoe, and all that happens to conservative (Hamiltonian) systems as they proceed toward chaos.

• Chaos in dynamical systems, Edward Ott, Cambridge(2002). Quite modern. Has a lot of interesting material on various sorts of fractal dimensions.

• Perspectives of nonlinear dynamics, Vols. 1 and 2, E. Atlee Jackson (Cambridge 1990)
  Interesting approaches to several things that are different from others. Toda waves, Lorenz system, many things in considerable detail with nice figures. It is definitely worth pursuing.

• Dynamical Systems, George D. Birkhoff (Amer. Math. Soc., 1927)
  An older place, often referenced, which told everything known in 1927.

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