Calculus of Variations I

Mariano Giaquinta, Stefan Hildebrandt

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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Contents

• I: 3
• II: 6
• III: 11
• V: 16
• VI: 27
• VII: 34
• VIII: 37
• XI: 43
• XII: 48
• XIII: 51
• XIV: 52
• XV: 55
• XVI: 59
• XVII: 68
• XVIII: 87
• XIX: 89
• XX: 97
• XXI: 110
• XXII: 122
• XXIII: 132
• XXIV: 145
• XXV: 147
• XXVI: 163
• XXVII: 172
• XXVIII: 182
• XXIX: 198
• XXX: 210
• XXXI: 217
• XXXII: 220
• XXXIII: 221
• XXXIV: 227
• XXXV: 229
• XXXVI: 232
• XXXVII: 236
• XXXVIII: 237
• XXXIX: 238
• XL: 242
• XLI: 250
• XLII: 251
• XLIII: 254
• XLIV: 260
• XLV: 264
• XLVI: 265
• XLVIII: 267
• XLIX: 271
• L: 276
• LI: 281
• LII: 286
• LIII: 292
• LIV: 306
• LV: 310
• LVI: 312
• LVII: 313
• LVIII: 327
• LIX: 332
• LX: 350
• LXI: 351
• LXII: 356
• LXIII: 362
• LXIV: 372
• LXV: 384
• LXVI: 395
• LXVII: 400
• LXX: 405
• LXXI: 408
• LXXII: 412
• LXXIII: 421
• LXXIV: 425
• LXXV: 432
• LXXVI: 437
• LXXVII: 468

Popular passages

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 463 - The Absolute Minimum in the Problem of the Surface of Revolution of Minimum Area...

Page 461 - Sufficient conditions by expansion methods for the problem of Bolza in the calculus of variations, Annals of Mathematics, (2), vol.

Page 461 - On the problem of Plateau. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 2.

Bibliographic information

• Title: Calculus of Variations I; Volume 310 of Grundlehren der mathematischen Wissenschaften
• Authors: Mariano Giaquinta, Stefan Hildebrandt
• Edition: illustrated, reprint
• Publisher: Springer Science & Business Media, 2004
• ISBN: 354050625X, 9783540506256
• Length: 474 pages
• Subjects: Mathematics / Calculus; Mathematics / Functional Analysis; Mathematics / General; Mathematics / Geometry / Differential; Science / Physics / Mathematical & Computational