N.A. Bobylov, S.V. Emel'yanov, S. Korovin

Geometrical Methods in Variational Problems

• Gebundenes Buch

Produktbeschreibung

Since the building of all the Universe is perfect and is cre ated by the wisdom Creator, nothing arises in the Universe in which one cannot see the sense of some maXImum or mInImUm Euler God moves the Universe along geometrical lines Plato Mathematical models of most closed physical systems are based on vari ational principles, i.e., it is postulated that equations describing the evolu tion of a system are the Euler~Lagrange equations of a certain functional. In this connection, variational methods are one of the basic tools for studying many problems of natural sciences. The first problems related to the search for extrema appeared as far back as in ancient mathematics. They go back to Archimedes, Appolonius, and Euclid. In many respects, the problems of seeking maxima and minima have stimulated the creation of differential calculus; the variational prin ciples of optics and mechanics, which were discovered in the seventeenth and eighteenth centuries, gave impetus to an intensive development of the calculus of variations. In one way or another, variational problems were of interest to such giants of natural sciences as Fermat, Newton, Descartes, Euler, Huygens, 1. Bernoulli, J. Bernoulli, Legendre, Jacobi, Kepler, La grange, and Weierstrass.

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Produktdetails

• Mathematics and Its Applications 485
• Verlag: Springer / Springer Netherlands
• Artikelnr. des Verlages: 978-0-7923-5780-3
• 1999.
• Seitenzahl: 564
• Erscheinungstermin: 31. Juli 1999
• Englisch
• Abmessung: 241mm x 160mm x 35mm
• Gewicht: 966g
• ISBN-13: 9780792357803
• ISBN-10: 0792357809
• Artikelnr.: 21566620

Inhaltsangabe

1 Preliminaries.- 1.1 Metric and Normed Spaces.- 1.2 Compactness.- 1.3 Linear Functional and Dual Spaces.- 1.4 Linear Operators.- 1.5 Nonlinear Operators and Functionals.- 1.6 Contraction Mapping Principle, Implicit Function Theorem, and Differential Equations on a Banach Space.- 2 Minimization of Nonlinear Functionals.- 2.1 Extrema of Smooth Functionals.- 2.2 Extremum of Lipschitzian and Convex Functionals.- 2.3 Weierstass Theorems.- 2.4 Monotonicity.- 2.5 Variational Principles.- 2.6 Additional Remarks.- 3 Homotopic Methods in Variational Problems.- 3.1 Deformations of Functionals on Hilbert Spaces.- 3.2 Deformations of Functionals on Banach Spaces.- 3.3 Global Deformations of Functionals.- 3.4 Deformation of Problems of the Calculus of Variations.- 3.5 Deformations of Lipschitzian Functions.- 3.6 Global Deformations of Lipschitzian Functions.- 3.7 Deformations of Mathematical Programming Problems.- 3.8 Deformations of Lipschitzian Functionals.- 3.9 Additional Remarks.- 4 Topological Characteristics of Extremals of Variational Problems.- 4.1 Smooth Manifolds and Differential Forms.- 4.2 Degree of Mapping.- 4.3 Rotation of Vector Fields in Finite-Dimensional Spaces.- 4.4 Vector Fields in Infinite-Dimensional Spaces.- 4.5 Computation of the Topological Index.- 4.6 Topological Index of Zero of an Isolated Minimum.- 4.7 Euler Characteristic and the Topological Index of an Isolated Critical Set.- 4.8 Topological Index of Extremals of Problems of the Calculus of Variations.- 4.9 Topological Index of Optimal Controls.- 4.10 Topological Characteristic s of Critical Points of Nonsmooth Functionals.- 4.11 Additional Remarks.- 5 Applications.- 5.1 Existence Theorems.- 5.2 Bounds of the Number of Solutions to Variational Problems.- 5.3 Applications of the Homotopic Method.- 5.4 Study of Degenerate Extremals.- 5.5 Morse Lemmas.- 5.6 Well-Posedness of Variational Problems. Ulam Problem.- 5.7 Gradient Procedures.- 5.8 Bifurcation of Extremals of Variational Problems.- 5.9 Eigenvalues of Potential Operators.- 5.10 Additional Remarks.- Bibliographical Comments.- References.

Rezensionen

"... the book is a valuable contribution to the literature. It is well-written, self-contained and it has an extensive bibliography, especially with regard to the literature in the Russian language."
(Mathematical Reviews, 2001a)