This text provides a clear concise introduction to the calculus of variations. The introduction chapter provides a general sense of the subject through a discussion of several classical and contemporary examples of the subject's use. In the second chapter the all-important Euler differential is dereived and the Lemma of Du Bois-Reymond is established. Other basic terminology is introduced and the underlying theory, which will later be extended smoothly to more complex problems, is developed. Chapters 3 through 6 are devoted to the crucial topics of necessary and sufficient conditions. The remainder of the text extends the concepts of the first half of hte book to problems involving variable boundaries, to parametic representations to the problems constrained by side conditions.
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Für höhere Schule und Studium
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ISBN-13
978-0-412-36700-7 (9780412367007)
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Schweitzer Klassifikation
Autor*in
University of Wurzburg, Darmstadt, Germany
Part 1 Introduction to the problem: examples; definition of the most important concepts; the question of the existence of solutions. Part 2 The Euler differential equation: derivation of the Euler differential equation; integration of the Euler equation under special assumptions on the function f; further examples and problems; new admissibility conditions and the lemma of Du Bois-Reymond; the Erdmann Corner conditions. Part 3 Sufficient conditions for variational problems with convex integrands: convex functions; a sufficient condition. Part 4 The necessary conditions of Weierstrass and Legendre: the Weierstrass necessary conditions; the Legendre necessary condition; examples and problems. Part 5 The necessary condition of Jacobi: the second variation and the accessory problem; Jacobi's necessary condition; one-parameter families of extremals and the geometric significance of conjugate points; concluding remarks about conjugate points. Part 6 The Hilbert independence integral and sufficiency conditions: fields of extremals; the Hilbert integral and the Weierstrass sufficient condition; further sufficient conditions; proof of the path-independence of the Hilbert integral. Part 7 Variational problems with variable boundaries: problems having a free boundary point; variational problems with boundaries a point and a curve; sufficient conditions for problems with moveable endpoints. Part 8 Variational problems depending of several functions: conditions of the first variation; further necessary and sufficient conditions. Part 9 The parametric problem: statement of the problem; necessary and sufficient conditions; some particulars of the parametric problem and proof of the second Erdmann Corner condition. Part 10 Variational problems with multiple integrals: statement of hte problem and examples; the Euler differential equations; sufficient conditions. Part 11 Variational problems with side conditions: the Lagrange multiplier rule; the isoperimetric problem; variational problems having an equation as a side condition; Lagrange's problem. Part 12 Introduction to direct methods.
