Lectures on Cauchy's Problem in Linear Partial Differential Equations

Description

Would well repay study by most theoretical physicists." — Physics Today

This book is a reprint of a volume, originally published by the Yale University Press, of Hadamard's lectures on hyperbolic differential equations, given at Yale in 1921. It is useful to have this fundamental analysis of the relation between equation type, boundary conditions, and solutions again generally available. Since the advent of quantum theory the interest in solutions of a hyperbolic differential equation (such as the wave equation) has rather changed its emphasis away from the study of the effects of boundary conditions on specified surfaces. Nevertheless, the results of such classical studies are still of interest and some of the techniques used (such as that of the Green's function) have continuing, and even enhanced, utility. The work reprinted here is one place, in the knowledge of the reviewer, where the effect of boundary conditions on the solution of hyperbolic equations is given in unified and general manner. Here is discussed, in some detail, the differences between hyperbolic and elliptic equations with respect to their characteristic surfaces, and the bearing this has on their differences with respect to boundary conditions. Here also are discussed, in considerable detail, the differences in behavior of waves in one, three, etc. dimensions and of waves in two, four, etc. dimensions. The techniques used in the analyses and in the exposition are not often those used in modern theoretical physics. Nevertheless, the book is one which would well repay study by most theoretical physicists.

—Philip M. Morse Massachusetts Institute of Technology

"An overwhelming influence on subsequent work on the wave equation." — Science Progress
"One of the classical treatises on hyperbolic equations." — Royal Naval Scientific Service

Delivered at Columbia University and the Universities of Rome and Zürich, these lectures represent a pioneering investigation. Jacques Hadamard based his research on prior studies by Riemann, Kirchhoff, and Volterra. He extended and improved Volterra's work, applying its theories relating to spherical and cylindrical waves to all normal hyperbolic equations instead of only to one. Topics include the general properties of Cauchy's problem, the fundamental formula and the elementary solution, equations with an odd number of independent variables, and equations with an even number of independent variables and the method of descent.

Hadamard's Pamphlets contains the paper "Les surfaces à courbures opposées et leurs lignes géodésiques" (p. 71; cf. Jaki's Uneasy Genius p. 350fn113), a classic in chaos theory, that inspired "Duhem's bull" (e.g., in his Aim & Structure of Physical Theory pp. 139 ff.).

cf. Park, Brett. “Blindspots of Empiricism in the Discovery of Chaos Theory.” Preprint, November 2025. https://philsci-archive.pitt.edu/27083/.