The Four Color Problem

The present volume represents the great majority of the papers presented on Sanibel Island at the first Symposium on Quantum Statistics and Many-Body Problems (January 26-29, 1975). In his Introductory Remarks, Professor Lawdin outlines the history of the original Symposia, and the genesis of the conference whose papers comprise this volume. We join him in his expression of thanks, and note, additionally, our gratitude to him and to Professors N. Y. Ohrn, J. R. Sabin, E. D. Adams, and John Daunt. The papers are grouped somewhat differently from their order of presentation. It seemed convenient to begin with the six papers which deal with sound propagation in one form or another, then have a two-paper diversion into solid Helium. The SHe superf1uid theme is picked up again with four papers on spin dynamics, orbit waves, etc., followed by a selection of five papers on a variety of experimental and theoretical aspects of the SHe superf1uid problem. Work in the areas of films, mono1ayers, and mixtures is presented next, followed by two papers on liquid ~He. We conclude with a selection of six papers on other quantum fluids and general statistical mechanics. We are most grateful to the contributors to this volume for their patience and cooperation; they have had as editors three utter novices! We have learned much, both scientifically and editorially. We hope that this volume will be of at least some help to others as well.

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. 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.

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