What Is the Lagrangian Counting?

Abstract

The calculus of variations provides an exhaustive descriptive account of the least-action principle and other icons of nature's parsimoniousness. However, when we seek explanations rather than mere formal descriptions of variational principles, almost invariably we discover a combinatorial origin; the best known examples are thermodynamics and Darwinian evolution. When it comes to the least-action principle, however, it is surprising that reductionistic attacks of this kind have been virtually absent. An eminent exception is, of course, Richard Feynman's explanation in terms of quantum path integrals, but even there, though the spirit of the approach is combinatorial, the nature of the objects that are “counted” is somewhat elusive; one is still explaining a mystery through another mystery. Feynman himself stresses that “whoever thinks they understand quantum mechanics, they don't,” and ultimately admits “I don't know what action is.” The challenge we proposes is to devise “classical” models (that is, models based on ordinary counting of large numbers of discrete objects rather than superposition of complex amplitudes) of classical analytical mechanics. Never mind what method nature actually uses; how come models of this kind—which, for example, were a dime a dozen for the laws of perfect gases—are so hard to come by for the least-action principle?