Joseph Ford

The Fermi-Pasta-Ulam problem: Paradox turns discovery☆

Abstract This pedagogical review is written as a personal retrospective which seeks to place the celebrated Fermi, Pasta, and Ulam paradox into historical perspective. After stating the Fermi-Pasta-Ulam results, we treat the questions it raises as a pedagogical “skeleton” upon which to drape (and motivate) the evolving story of nonlinear dynamics/chaos. This review is thus but another retelling of that story by one intimately involved in its unfolding. This is done without apology for two reasons. First, if my colleagues have taught me anything, it is that an audience of experts will seldom pay greater attention than when, with some modicum of grace and polish, they are told things they know perfectly well already. Second, if generations of students have taught me anything, it is that few things fascinate them more than a scientific mystery — and the Fermi-Pasta-Ulam paradox is a cracker-jack mystery. And so readers, especially graduate students curious about nonlinear dynamics/chaos, are now invited to sit back, loosen their belts (and minds), and prepare for fact that sometimes reads like fantasy.

COMPUTER STUDIES OF ENERGY SHARING AND ERGODICITY FOR NONLINEAR OSCILLATOR SYSTEMS

Weakly coupled systems of N oscillators are investigated using Hamiltonians of the form H=12 ∑ k=1N(pk2+ωk2qk2)+α ∑ j,k,l=1NAjklqjqkql, where the Ajkl are constants and where α is chosen to be sufficiently small that the coupling energy never exceeds some small fraction of the total energy. Starting from selected initial conditions, a computer is used to provide exact solutions to the equations of motion for systems of 2, 3, 5, and 15 oscillators. Various perturbation schemes are used to predict, interpret, and extend these computer results. In particular, it is demonstrated that these systems can share energy only if the uncoupled frequencies ωk satisfy resonance conditions of the form ∑ nkωk ≲α for certain integers nk determined by the particular coupling. It is shown that these systems have N normal modes, where a normal mode is defined as motion for which each oscillator moves with essentially constant amplitude and at a given frequency or some harmonic of this frequency. These systems are shown to ha...

Does quantum mechanics obey the correspondence principle? Is it complete?

This elementary review paper presents the compelling evidence which supports the notion that quantum mechanics is much too simple a theory to adequately describe a complex world. Rigorous arguments based on algorithmic complexity theory are used to show that both the quantum Arnol’d cat and a broad category of finite, bounded, undriven, quantum systems do not obey the correspondence principle, implying that quantum mechanics is also not complete. An experiment, well within current laboratory capability, is proposed which can expose the inability of quantum mechanics to adequately describe macroscopic chaos. In its final section, this paper describes a theoretical framework that provides a proper setting for interpreting these surprising results.

On the Stability of Periodic Orbits for Nonlinear Oscillator Systems in Regions Exhibiting Stochastic Behavior

A computer has been used to determine the stability character of periodic orbits for the Hamiltonian oscillator system H=12(p12+p22+q12+q22)+q12q2−13q23. Using procedures developed by Greene [J. Math. Phys. 9, 760 (1968)], empirical evidence has been obtained indicating that this system has a dense or near dense set of unstable periodic orbits throughout its stochastic (unstable) regions of phase space. The extent to which such stochastic regions exhibit C‐system behavior, i.e., ergodicity and mixing, is discussed. Finally, the above Hamiltonian system is shown to be intimately related to the Fermi‐Pasta‐Ulam system as well as to the Toda lattice.

CHAOS: SOLVING THE UNSOLVABLE, PREDICTING THE UNPREDICTABLE!

Publisher Summary This chapter focuses on chaos. Chaos means exponentially sensitive dependence of final state upon initial state, positive Liapunov exponents, positive topological or metric entropy, and fractal attractors. Chaos means deterministic randomness —deterministic because of existence–uniqueness and randomness. Chaos is a synonym for randomness. Almost all infinite binary sequences, having positive complexity, pass every computable test for randomness, where a computable test is one expressible as a finite algorithm. A-integrable systems can approximate randomness in the same sense that rationals can approximate irrationals. The virtue of plane billiards, as a class, is that they are the simplest systems that exhibit almost all possible dynamical behavior.

Theory of Ultrasonic Pulse Measurements of Third‐Order Elastic Constants for Cubic Crystals

The equations of motion for an elastic nonisotropic solid are reduced to a form useful for determination of third‐order elastic constants by means of ultrasonic pulse distortion measurements. Values of the coefficients in the reduced equations of motion are tabulated for two cubic materials. The tables are used to show that for cubic materials one should be able to measure C111, C112, and C166 with reasonable accuracy. An argument is given which shows that the equations of motion for a single plane wave in a cubic crystal depend on the five parameters C111, C112, C166, (2C144+C123), and (½C144+C456) instead of all six third‐order elastic constants.

Numerical study of billiard motion in an annulus bounded by non-concentric circles

Abstract This paper exposes a simple, focusing-dispersing billiard system whose phase space simultaneously exhibits the full range of possible Hamiltonian syste, orbit behavior over a continuous range of system parameter values. Specifically, we find that billiard motion between two non-concentric circles is characterized by three distinct phase space regions: (1) a rigorously integrable region in which billiard orbits undergo collisions only with the outer circle boundary; (2) a KAM near-integrable region in which billiard collisions strictly alternate between inner and outer circles; and (3) a chaotic region produced by orbital sequences of billiard collisions which randomly alternate between the integrable and near-integrable patterns. The properties of the integrable, near-integrable, and chaotic regions, the presence of island chains, and the homoclinic points associated with certain hyperbolic fixed points are discussed. Perhaps the most interesting feature of this billiard system, however, is the fresh view of the source of chaos it provides; specifically, the chaotic orbits of region (3) exhibit “random” jumping between the extended invariant curves of regions (1) and (2).

Analytically solvable dynamical systems which are not integrable

Abstract The central, two-fold aim of this paper is to expose a class of nonintegrable systems whose solutions can be analytically expressed in terms of computationally tractable algorithms and to explicitly construct a demonstrably meaningful general solution for one member of this class. Toward this end, information theoretic results are reviewed which show that dynamical systems of null metric entropy have general solutions of null algorithmic complexity. Hence, all systems exhibiting null metric entropy, including ergodic (only) and mixing (only) systems, have orbits which can be specified by algorithmically meaningful, analytic expressions whose information content is very much less than that of the orbits they describe. In consequence, we may quite legitimately define the class of systems having null metric entropy ot be algorithmically integrable, called A-integrable herein. After establishing the existence of A-integrability, this paper turns to the problem of exposing a specific nonintegrable but nonetheless A-integrable system whose algorithmically meaningful solution can be explicitly derived. The billiard moving within a plane polygon having the shape of a 60°–120° rhombus is shown to be a simple example. Indeed, the whole class of billiards moving in polygons whose angles are all rational multiples of π have null metric entropy and are therefore A-integrable systems; moreover, all rational billiards appear to have explicitly derivable solutions as is discussed herein and in a recent preprint by Bjorn Birnir. Because rational billiards are dense in the set of all plane billiards, including chaotic ones, the present paper suggests that A-integrable billiards may provide a long sought analytic route to chaos. Finally, it is pointed out that null entropied, A-integrable systems are perhaps the broadest category of systems which can reasonably be termed integrable, for systems having positive metric entropy are truly nonintegrable due to the presence of orbits which are their own simplest description and their own fastest computer.

Marginal local instability of quasi-periodic motion☆

Abstract In this paper, we analytically prove a long suspected link between integrable hamiltonian systems and average linear growth with time of separation distance between initially close phase space states. Specifically, it is shown that almost all solutions to the linearized variational equations derived from bounded, integrable hamiltonian systems exhibit an average linear growth with time, becoming unbounded at t →∞. The orbits of bounded, integrable hamiltonian systems are thus always locally marginally unstable, forever lying on that sharp border which divides completely stable from completely unstable motion.

Theory of Ultrasonic Three‐Phonon Interactions in Single‐Crystal Solids

We derive expressions relating wave amplitudes and third‐order constants for resonant ultrasonic three‐phonon interactions in single‐crystal solids. Ten resonant interactions are tabulated for the case of copper. We reemphasize earlier suggestions that three‐phonon interactions can be used to determine third‐order elastic constants.