Calculus of Variations

ISBN-10: 0486630692

ISBN-13: 9780486630694

Edition: 1974

Authors: Robert Weinstock

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Book details

List price: $18.95
Copyright year: 1974
Publisher: Dover Publications, Incorporated
Publication date: 6/1/1974
Binding: Paperback
Pages: 352
Size: 5.75" wide x 8.75" long x 0.75" tall
Weight: 1.144
Language: English

ROBERT WEINSTOCK is a graphic designer, former children's book editor, and author of Gordimer Byrd's Reminder. He lives in New York City.

Background Preliminaries
Piecewise continuity, piecewise differentiability
Partial and total differentiation
Differentiation of an integral
Integration by parts
Euler's theorem on homogeneous functions
Method of undetermined lagrange multipliers
The line integral
Formula for surface area
Taylor's theorem for functions of several variables
The surface integral
Gradient, laplacian
Green's theorem (two dimensions)
Green's theorem (three dimensions)
Introductory Problems
A basic lemma
Statement and formulation of several problems
The Euler-Lagrange equation
First integrals of the Euler-Lagrange equation
A degenerate case
The brachistochrone
Minimum surface of revolution
Several dependent variables
Parametric representation
Undetermined end points
Brachistochrone from a given curve to a fixed point
Isoperimetric Problems
The simple isoperimetric problem
Direct extensions
Problem of the maximum enclosed area
Shape of a hanging rope
Restrictions imposed through finite or differential equations
Geometrical Optics: Fermat's Principle
Law of refraction (Snell's law)
Fermat's principle and the calculus of variations
Dynamics of Particles
Potential and kinetic energies
Generalized coordinates
Hamilton's principle
Lagrange equations of motion
Generalized momenta
Hamilton equations of motion
Canonical transformations
The Hamilton-Jacobi differential equation
Principle of least action
The extended Hamilton's principle
Two Independent Variables: The Vibrating String
Extremization of a double integral
The vibrating string
Eigenvalue-eigenfunction problem for the vibrating string
Eigenfunction expansion of arbitrary functions
Minimum characterization of the eigenvalue-eigenfunction problem
General solution of the vibrating-string equation
Approximation of the vibrating-string eigenvalues and eigenfunctions (Ritz method)
Remarks on the distinction between imposed and free end-point conditions
The Sturm-Liouville Eigenvalue-Eigenfunction Problem
Isoperimetric problem leading to a Sturm-Liouville system
Transformation of a Sturm-Liouville system
Two singular cases: Laguerre polynomials, Bessel functions
Several Independent Variables: The Vibrating Membrane
Extremization of a multiple integral
Change of independent variables
Transformation of the laplacian
The vibrating membrane
Eigenvalue-eigenfunction problem for the membrane
Membrane with boundary held elastically
The free membrane
Orthogonality of the eigenfunctions
Expansion of arbitrary functions
General solution of the membrane equation
The rectangular membrane of uniform density
The minimum characterization of the membrane eigenvalues
Consequences of the minimum characterization of the membrane eigenvalues
The maximum-minimum characterization of the membrane eigenvalues
The asymptotic distribution of the membrane eigenvalues
Approximation of the membrane eigenvalues
Theory of Elasticity
Stress and strain
General equations of motion and equilibrium
General aspects of the approach to certain dynamical problems
Bending of a cylindrical bar by couples
Transverse vibrations of a bar
The eigenvalue-eigenfunction problem for the vibrating bar
Bending of a rectangular plate by couples
Transverse vibrations of a thin plate
The eigenvalue-eigenfunction problem for the vibrating plate
The rectangular plate
Ritz method of approximation
Quantum Mechanics
First derivation of the Schr�dinger equation for a single particle
The wave character of a particle. Sec
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