Exterior Differential Systems and the Calculus of Variations

This monograph is a revised and expanded version of lecture notes from a class given at Harvard University, Nankai University, and the Graduate School of the Academia Sinica during the academic year 1981-82. The objective was to present the formalism, together with numerous illustrative examples, of the calculus of variations for functionals whose domain of definition consists of integral manifolds of an exterior differential system. This includes as a special case the Lagrange problem of analyzing classical functionals with arbitrary (i.e., nonholonomic as well as holonomic) constraints. A secondary objective was to illustrate in practice some aspects of the theory of exterior differential systems. This includes as a special case the Lagrange problem of analyzing classical functionals with arbitrary (i.e., nonholonomic as well as holonomic) constraints. A secondary objective was to illustrate in practice some aspects of the theory of exterior differential systems. In fact, even though the calculus of variations is a venerable subject about which it is hard to say something new, (l) we feel that utilizing techniques from exterior differential systems such as Cauchy characteristics, the derived flag, and prolongation allows a systematic treatment of the subject in greater generality than customary and sheds new light on even the classical Lagrange problem.
As indicated by the table of contents the text is divided into four chapters, with most of the general theory being presented in the first and last. We break somewhat with current tradition in that an unusually large amount of space is devoted to examples. Perhaps even more of a break (or is it a regression?) is the special concern given to the explicit integration of the Euler-Lagrange equations, Jacobi equations, Hamilton-Jacobi equations, etc. in these examples-in a word we want to get out formulas. Much of the middle two chapters are devoted to methods for doing this; again the theory of exterior differential systems provides an effective computational tool. (2)


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