Calculus of Variations I

Mariano Giaquinta, Stefan Hildebrandt (Autoren)

Buch | Softcover

XXIX, 474 Seiten

2010 | 1. Softcover reprint of hardcover 1st ed. 1996
Springer Berlin (Verlag)
978-3-642-08074-6 (ISBN)

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This book describes the classical aspects of the variational calculus which are of interest to analysts, geometers and physicists alike. Volume 1 deals with the for mal apparatus of the variational calculus and with nonparametric field theory, whereas Volume 2 treats parametric variational problems as well as Hamilton Jacobi theory and the classical theory of partial differential equations of first ordel;. In a subsequent treatise we shall describe developments arising from Hilbert's 19th and 20th problems, especially direct methods and regularity theory. Of the classical variational calculus we have particularly emphasized the often neglected theory of inner variations, i. e. of variations of the independent variables, which is a source of useful information such as mono tonicity for mulas, conformality relations and conservation laws. The combined variation of dependent and independent variables leads to the general conservation laws of Emmy Noether, an important tool in exploiting symmetries. Other parts of this volume deal with Legendre-Jacobi theory and with field theories. In particular we give a detailed presentation of one-dimensional field theory for nonpara metric and parametric integrals and its relations to Hamilton-Jacobi theory, geometrical optics and point mechanics. Moreover we discuss various ways of exploiting the notion of convexity in the calculus of variations, and field theory is certainly the most subtle method to make use of convexity. We also stress the usefulness of the concept of a null Lagrangian which plays an important role in we give an exposition of Hamilton-Jacobi several instances.

of Calculus of Variations I.- 1. The First Variation.- 2. Variational Problems with Subsidiary Conditions.- 3. General Variational Formulas.- 4. Second Variation, Excess Function, Convexity.- 5. Weak Minimizers and Jacobi Theory.- 6. Weierstrass Field Theory for One-Dimensional Integrals and Strong Minimizers.- Supplement. Some Facts from Differential Geometry and Analysis.- 1. Euclidean Spaces.- 2. Some Function Classes.- 3. Vector and Covector Fields. Transformation Rules.- 4. Differential Forms.- 6. Mean Curvature and Gauss Curvature.

Erscheint lt. Verlag	4.12.2010
Reihe/Serie	Grundlehren der mathematischen Wissenschaften
Zusatzinfo	XXIX, 474 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	748 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
	Naturwissenschaften ► Physik / Astronomie
Schlagworte	Calculus • Calculus of Variations • Convexity • Curvature • differential equation • Differential Geometry • Gaussian curvature • Hamiltonian Formalism • Lagrangian Formalism • mean curvature • Vector field
ISBN-10	3-642-08074-X / 364208074X
ISBN-13	978-3-642-08074-6 / 9783642080746
Zustand	Neuware