Hamiltonian Chaos

Cartesian coordinates, generalized coordinates, canonical coordinates, and, if you can solve the problem, action-angle coordinates. That is not a sentence, but it is classical mechanics in a nutshell. You did mechanics in Cartesian coordinates in introductory physics, probably learned generalized coordinates in your junior year, went on to graduate school to hear about canonical coordinates, and were shown how to solve a Hamiltonian problem by finding the action-angle coordinates. Perhaps you saw the action-angle coordinates exhibited for the harmonic oscillator, and were left with the impression that you (or somebody) could find them for any problem. Well, you now do not have to feel badly if you cannot find them. They probably do not exist! Laplace said, standing on Newton’s shoulders, “Tell me the force and where we are, and I will predict the future!” That claim translates into an important theorem about differential equations—the uniqueness of solutions for given initial conditions. It turned out to be an elusive claim, but it was not until more than 150 years after Laplace that this elusiveness was fully appreciated. In fact, we are still in the process of learning to concede that the proven existence of a solution does not guarantee that we can actually determine that solution. In other words, deterministic time evolution does not guarantee predictability. Deterministic unpredictability or deterministic randomness is the essence of chaos. Mechanical systems whose equations of motion show symptoms of this disease are termed nonintegrable. Nonintegrability is not the result of insufficient brainpower or inadequate computational power. It is an intrinsic property of most nonlinear differential equations with three or more variables. In principle, Newton’s laws can predict the indefinite future of a mechanical system. But the distant future of a nonintegrable system must be “discovered” by numerical integration, one time step after another. The further into the future that prediction is to be made, or the more precise it is to be, the more precise must be our knowledge of the initial conditions, and the more precise must be the numerical integration procedure. For chaotic systems the necessary precision (for example, the number of digits to be retained) increases exponentially with time. The practical limit that this precision imposes on predictions made by real, necessarily finite, computation was emphasized in a recent article in this column. In the Hamiltonian formulation of classical dynamics, a system is described by a pair of first-order ordinary differential equations for each degree of freedom i. The dynamical variables are a canonical coordinate qi and its conjugate