Calculus of Variations I
Springer Science & Business Media, Jun 23, 2004 - Mathematics - 474 pages
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.
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arbitrary assume assumption boundary bundle calculus of variations called Chapter closed condition conjugate connected consider constant contained continuous convex corresponding curvature curve defined definition denotes dependent derive described determined discussion domain Du(x equivalent Euler equations example exists expression extremal fact field fixed follows formula function geodesic geometric given graph Hence holds implies independent infer inner integral introduce invariant Jacobi Lagrange Lagrangian Lemma linear mapping Math Mayer field means minimizer Moreover motion multiplier necessary Note null Lagrangian obtain operator particular positive principle problem Proof Proposition proved reasoning regular relation respect result satisfies smooth solution space strong sufficient Suppose surface theorem theory transformation transversality variables variational integral vector weak minimizer whence write yields
Page 464 - RESEARCHES IN THE CALCULUS OF VARIATIONS, principally on the Theory of Discontinuous Solutions: an Essay to which the Adams' Prize was awarded in the University of Cambridge in 1871.
Page 461 - Sufficient conditions by expansion methods for the problem of Bolza in the calculus of variations, Annals of Mathematics, (2), vol.
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