Calculus of Variations I

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Springer Science & Business Media, Jun 23, 2004 - Mathematics - 474 pages
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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.
 

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Contents

I
3
II
6
III
11
V
16
VI
27
VII
34
VIII
37
XI
43
XL
242
XLI
250
XLII
251
XLIII
254
XLIV
260
XLV
264
XLVI
265
XLVIII
267

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
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
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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.

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