Calculus of Variations

Contents: 1. The first variation of the integral. 2. The second variation of the integral. 3. Sufficient conditions for an extremum of the integral. 4. Weierstrass s theory of the problem in parameter representation. 5. Kneser s theory. 6. Weierstrass s theory of the isoperimetric problems. 7. Hilbert s existence theorem. Index. An understanding of variational methods, the source of such fundamental theorems as the principle of least action and its various generalizations, is essential to the study of mathematical physics and applied mathematics. In this highly regarded text, the author develops the calculus of variations both for its own intrinsic interest and because of its wide and powerful applications to modern mathematical physics. The purpose of this text is to lay a broad foundation for an understanding of the problems of the calculus of variations and its many methods and techniques, and to prepare readers for the study of modern optimal control theory. The treatment is limited to a thorough discussion of single-integral problems in one or more unknown functions, where the integral is problems in one or more unknown functions, where the integral is employed in the Riemannian sense. This book, which includes many strategically placed problems, is directed to advanced undergraduate and graduate students with a background in advanced calculus and intermediate differential equations, and is adaptable to either a one-or two semester course on the subject.…