Theprincipiaand contemporary mechanics: Chaotic dynamics and the new unpredictability

Abstract

The legacy of Sir Isaac Newton’sPrincipiais evident today throughout the mathematical sciences. Applications of his mechanics range from protein dynamics, through macroscopic engineering and space flight, to pattern formation in galactic super-clusters, and beyond mechanics the Newtonian paradigm of differential evolution has blossomed into biology, ecology and economics. Computers have naturally had a profound impact, making long, precise time integrations a routine facility. They have, moreover, played a seminal role in the revolutionary new theory of chaotic motions. These unexpected, noisy motions arise in well posed, deterministic systems of differential equations, a recent illustration from conservative Hamiltonian dynamics being the chaotic tumbling of the Saturnian satellite, Hyperion. They are typical responses of nonlinear dissipative models once the phase dimension exceeds two: Lorenz’s historic study of turbulent atmospheric convection, and fractal escape boundaries of a potential well, are here used to illustrate salient features. Of particular philosophical significance is the exponential divergence from adjacent starts, whose finite precision implies a definite time-horizon, beyond which prediction is impossible. Coupled with the repetitive mixing of the chaotic flows, this divergence dictates that macroscopic features (such as the occurrence of clockwise as opposed to anticlockwise convective rolling in the Lorenz model) can occur in any sequence, and in this respect the motions are as random as a coin toss.