Calculus of Variations ISpringer Science & Business Media, 23.06.2004 - 474 Seiten 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. |
Inhalt
I | xxxiii |
II | xxxiii |
III | xxxiii |
V | 16 |
VI | 27 |
VII | 34 |
VIII | 37 |
XII | 43 |
XLI | 242 |
XLII | 250 |
XLIII | 251 |
XLIV | 254 |
XLV | 260 |
XLVI | 264 |
XLVII | 265 |
XLIX | 267 |
XIII | 48 |
XIV | 51 |
XV | 52 |
XVI | 55 |
XVII | 59 |
XVIII | 68 |
XIX | 87 |
XX | 89 |
XXI | 97 |
XXII | 110 |
XXIII | 122 |
XXIV | 132 |
XXV | 145 |
XXVI | 147 |
XXVII | 163 |
XXVIII | 172 |
XXIX | 182 |
XXX | 198 |
XXXI | 210 |
XXXII | 217 |
XXXIII | 220 |
XXXIV | 221 |
XXXV | 227 |
XXXVI | 229 |
XXXVII | 232 |
XXXVIII | 236 |
XXXIX | 237 |
XL | 238 |
L | 271 |
LI | 276 |
LII | 281 |
LIII | 286 |
LIV | 292 |
LV | 306 |
LVI | 310 |
LVII | 312 |
LVIII | 313 |
LIX | 327 |
LX | 332 |
LXI | 350 |
LXII | 351 |
LXIII | 356 |
LXIV | 362 |
LXV | 372 |
LXVI | 384 |
LXVII | 395 |
LXVIII | 400 |
LXIX | 405 |
LXX | 408 |
LXXI | 412 |
LXXII | 421 |
LXXIII | 425 |
LXXIV | 432 |
437 | |
468 | |
Häufige Begriffe und Wortgruppen
assume boundary conditions calculus of variations Carathéodory catenary class C¹ conjugate point conjugate value consider convex curvature curve defined denotes derive diffeomorphism differential equations Dirichlet integral domain Du(x dx¹ eigenvalue eikonal equivalent Euler equations Euler operator extremal field extremal of F F₂ field f field theory follows formula function geodesic given graph Hence holds implies infer inner extremal integral F(u isoperimetric problem Jacobi fields Lagrange Lagrange problem Lemma Lp(u mapping Mayer field mean curvature minimizer of F Noether's Noether's theorem null Lagrangian obtain one-dimensional p-form P₂ parameter Proof Proposition respect satisfies solution strong minimizer sufficient conditions Suppose surface tangent theorem transformation transversality v₁ variational integral variational problems vector field weak extremal weak minimizer Weierstrass whence x₁ z₁
Beliebte Passagen
Seite 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.
Seite 463 - The Absolute Minimum in the Problem of the Surface of Revolution of Minimum Area...
Verweise auf dieses Buch
Nonlinear Functional Analysis and Its Applications: Part 2 B: Nonlinear ... E. Zeidler Keine Leseprobe verfügbar - 1989 |