Introduction for
Nonlinear Mechanics
Spring, 2004

Material about textbooks now moved from homepage to the bottom of these pages.
Below that there is a partial syllabus for the course, originally also on the homepage.

There was a time, perhaps 45-60 years ago, when classical mechanics was considered a reasonably boring subject, that students must be subjected to, but which, at most, only served as a foundation for more interesting, and more modern, subfields of physics. This time has, happily passed, and the subject, now often referred to by some phrase which includes the word "nonlinear" within it, is the subject of large quantities of literature, and, as a field, is a very long ways from even being well understood. One might even wonder why it was that the subject really went "out of fashion" for a relatively long period of time. Perhaps one reason is that it was finally understood that the (remaining) unsolved problems were in fact very difficult: the work of Poincaré and others, a bit before the end of the 19th century, made this very clear. Prior to that time a great deal of work had been done on problems in mechanics, especially the several-body problem applicable to the solar system, with perturbation methods, that gave results that often agreed at least rather well with observations. On the other hand, Poincaré's work on questions of very long term stability caused some re-direction of thought, toward new approaches and the need for new mathematics to handle them. The belief that such questions could, eventually, be answered caused a lot of effort to be put into these problems, with the expectation that one would, again eventually, be able to predict stability of bounded orbits, which is some sort of generalization of periodicity. Such ideas might, perhaps, be classed loosely under the rubric of integrability, taken from Liouville's descriptions of the ways in which a system could, at least in principle, be reduced to one where the "answers" could be found. From a very different point of view such statements about stability of bounded orbits can be thought of as statements that say that rather small regions of the theoretically-possible volume of phasespace would be needed for any single such orbit. This attitude was, however, contradictory to the efforts of some other groups of physicists who wanted to show that classical statistical mechanics followed (easily ?) from the classical mechanics of single particles when one considered sufficiently large ensembles of particles. More precisely the ergodic hypothesis, as such a notion is called when it is more carefully formulated, would claim that most motions of large numbers of particles would be random, so that (at least almost) the entirety of phasespace would be explored by objects that moved in a sufficiently complicated dynamical situation, such as molecules in a real gas. In fact, it would seem that it was the majority view of most physicists in the times I mentioned above, namely when mechanics was more or less out of favor, that somehow one could believe both these views at once, with, of course, the exception of a very few who were still actively involved in trying to understand them better. However, two "new" things appeared on the scene which changed the situation fairly drastically: one of these was the possibility of doing really serious numerical experiments, on large, fast computers; the other was the creation, and application to problems in mechanics, of rather new, more powerful, and more esoteric, branches of mathematics. In particular the so-called Fermi-Pasta-Ulam problem was a very interesting one, where these three researchers had the availability of some quite new computers (at Los Alamos), and decided to show that the ergodic hypothesis was valid for a 1-dimensional system of particles (on a string, if you like) that interacted with each other in a nonlinear way so that it was not really susceptible to analytic methods. They inserted some energy in a particular excited mode of the system, put it into the computer, and allowed it to work away, expecting that this energy, over time, would be distributed among all the modes of the system in a way appropriate to the excitation energies of the various levels, i.e., in much the same way that this does happen in ideal gases and their reasonable generalizations. This is simply a particular statement of the ergodic hypothesis for this problem. However, in fact they were quite disappointed to find, instead, periodic recurrences of various sorts. Historians suggest that it was actually this particular nonlinear problem, experimentally resolved on a sufficiently powerful computer, that began the impetus that has by now led to a very large body of new methods to deal with nonlinear ode's and pde's that describe dynamical problems of many sorts, leading to soliton methods, inverse scattering problems, etc. On the other hand, it was true that statistical mechanics often worked very well, even though no general proofs of the ergodic hypothesis had been devised at that time. Nowadays, there are in fact some rather specific proofs of such a thing, but, to the best of my knowledge, often in the actual limit of systems with infinitely many particles. It must therefore, somehow or other, be true that some given dynamical system can have both sorts of behavior, perhaps even under very "nearby" sorts of conditions. On the other hand, one may also say that there are in fact solutions to the questions surrounding this ergodic hypothesis, that are given by the (celebrated) KAM theorem, due to Kolmogorov, Arnold, and Moser. Given a dynamical system that is "not too far away" from a system that can be understood, this theorem provides tools to answer the question of whether trajectories of some particular dynamical system does in fact, or not, travel through the entire region of the phase space that would be (energetically) allowed to it; i.e., to calculate the topology of that part of the phase space through which trajectories travel. Their efforts occurred in the 1950's and 1960's, although an understanding of the relevance of the theorem did take quite a few more years to "percolate" into the awareness of the majority of physicists. This then did provide an important topological tool for studying mechanics about the same time as the understandings of new methods of resolving nonlinear pde's emerged, beginning, perhaps with the FPU problem, the KdV problem, and the sine-Gordon equation, and, also the introduction of powerful computers to do detailed numerical experiments. The coming together of all these things, it would seem, caused a resurgence of interest in what is now often called nonlinear dynamics. These days our understanding of mechanics is somewhat different from the views in, say, the first edition of Goldstein's text. It is understood that in fact the vast majority of problems in mechanics are not solvable, but there are in fact ways to study in detail many particular ones that are of interest, and to answer many of the questions that one might want answered. We want to spend time this semester trying to get some "feel" for what can, and cannot, be done, and how to do it. A very basic idea is then to take some given dynamical system, with sets of initial conditions and operational parameters, and try to divine the values for which it has periodic, or recurrent, behavior, those for which its behavior is chaotic, and how the division between these behaviors is defined. As well it is important to understand the situations where these behaviors are stable or unstable, with respect to not-necessarily-too-small perturbations, since we can probably not get exact answers to most of the interesting questions, even numerically. This definitely requires us to also divide problems into those which are conservative (or Hamiltonian) and those which are dissipative. For one-dimensional problems we can provide a good answer to these questions; for two-dimensional (autonomous) problems the methods at least are clear, where autonomous means that the problem has no explicit time dependence. However, even here the phasespace is 4-dimensional, so that explicit graphs are often difficult to produce or interpret. Here the introduction of Poincaré sections will provide more information than might have a priori been expected. These sections will then lead us into the consideration of (discrete) maps, which are often somewhat easier to analyze. For more degrees of freedom than that the routes to finding answers to problems get hazier, and we will surely spend much less time there. The goal should be to understand, and determine, various important and useful properties of simple dynamical systems: periodicity, bifurcations, chaotic behavior, etc. As well, although we will make no attempt to "prove" the KAM theorem, we would like to understand it well enough to get useful information out of it for these systems. This will lead us to better understand the ideas behind, for instance, integrability, Lyapunov exponents, universal constants, KAM surfaces, limit cycles, strange attractors, and probably a number of other ideas that don't occur to me at the moment. As described in the history above, the method will oscillate back and forth between numerical and analytic, and will seldom venture further than 4-dimensional phasespaces. Since we will usually only be able to consider a system approximately, we will spend a fair amount of time on various variants of perturbation theory, almost all of which is based on Hamiltonian dynamics, and, in particular, the Hamilton-Jacobi (action/angle variable) approach to it.

(Optional) Textbooks, and (Partial) Syllabus