Calculus of Variations and Partial Differential Equations of the First Order

Description

cited in this lecture by Stefan Hildebrandt

Hildenbrandt quotes Carathéodory as saying that Lie's contact transformations are a useless aparatus ☺

W. R. Hamilton, and shortly afterwards G. G. Jacobi, discovered the connection between partial differential equations of the first order and the ordinary differential equations of mechanics or geometrical optics or, generally, the ordinary differential equations of any variational problem for functions of one variable. Thus was initiated one of the most fruitful developments of analysis in the 19th century. The key to this "Hamilton-Jacobi theory'' is the fact that the characteristics of the partial differential equations are identical with the trajectories of the ordinary differential equations of the corresponding variational problem. The integration of these "canonical'' systems of ordinary differential equations and the corresponding partial differential equation are equivalent problems, and a great wealth of practical applications and theoretical insights emerged. Later in the 19th century, many mathematicians, such as Weierstrass and Lie, made important further contributions; even features of the modern theory of differential forms are related to those fields.
The original work by Hamilton and Jacobi, although tied to questions of mathematical physics, was presented in a style of formal elegant algorithm. Then in 1900, D. Hilbert, on the basis of the invariant character of the variational integral, brushed these formal aspects aside and greatly clarified the geometric intuitive aspects inherent in the original motivation by geometrical optics (Fermat's principle) or mechanics (Hamilton's principle).
The famous German book by the author, which appeared first in 1935 [Variationsrechnung und partielle Differential-gleichungen erster Ordnung, Teubner, Leipzig, 1935; second edition, Band I: Theorie der partiellen Differential-gleichungen erster Ordnung, 1956; MR0089338], was a successful attempt at reconciling the old formal approach and the newer attitude of a conceptual penetration of the subject. The original book is a masterpiece of mathematical writing. It is gratifying that with the important personal cooperation of E. Hölder as editor, an American translation is appearing which does full justice to the literary qualities of the original and which moreover contains, and promises, most valuable comments and supplementations by E. Hölder, including a third supplementary volume.
The present, relatively short, first volume is mostly restricted to elementary, introductory material and to the formal theory of partial differential equations of the first order and related subjects, such as the techniques of "Poisson's brackets'' and similar symbols, to canonical and other contact transformations, Pfaff's problems, etc. Only in the forthcoming second volume will the more varied and colorful aspects of the calculus of variations proper be presented, and the third volume by E. Hölder will concern, among other topics, the links between the classical theory and modern theories, such as the theory of differential forms. The total work, when available, must be expected to be a most stimulating contribution to the mathematical literature.

Reviewed by R. Courant