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Calculus of Variations: With Applications to Physics and Engineering

Robert Weinstock
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is an introduction to the calculus of variations that concentrates very heavily on its applications, and only outlines the general theory. It makes heavy use of eigenfunctions and does have a good discussion of those. The present volume is a Dover 1974 corrected reprint of the 1952 McGraw-Hill edition.

The book has good coverage of all the classical problems: geodesics, brachistochrone, isoperimetric problems, and Snell’s law (refraction of light). It also deals with a large number of oscillation (vibration) problems in mechanics and in quantum theory. There’s also some treatment of electrostatics and elasticity. The organizing principles for these physical problems are Hamiltonians and the principle of least action. The treatment is interesting because it develops the partial differential equations for these problems afresh from variational principles each time rather than referring to the general wave equation.

The exercises are numerous and very good, but focus (like the rest of the book) on particular physical problems.

This is still an excellent book for applications, although not as good for math students. Another classical and more mathematical book is Gelfand & Fomin’s Calculus of Variations (Dover 2000 reprint of the 1963 edition). Some more recent and more mathematical books that are well-regarded are Mark Kot’s A First Course in the Calculus of Variations, Bernard Dacorogna’s Introduction to the Calculus of Variations, and Bruce van Brunt’s The Calculus of Variations. There are good quick introductions to the classical calculus of variations in Mary L. Boas’s Mathematical Methods in the Physical Sciences (Chapter 9) and Arfken et al.’s Mathematical Methods for Physicists (Chapter 22).

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences


Chapter 1. Introduction
Chapter 2. Background Preliminaries
    1. Piecewise continuity, piecewise differentiability 2. Partial and total differentiation 3. Differentiation of an integral 4. Integration by parts 5. Euler's theorem on homogeneous functions
    6. Method of undetermined lagrange multipliers 7. The line integral 8. Determinants 9. Formula for surface area 10. Taylor's theorem for functions of several variables
    11. The surface integral 12. Gradient, laplacian 13. Green's theorem (two dimensions) 14. Green's theorem (three dimensions)
Chapter 3. Introductory Problems
    1. A basic lemma 2. Statement and formulation of several problems 3. The Euler-Lagrange equation 4. First integrals of the Euler-Lagrange equation. A degenerate case 5. Geodesics
    6. The brachistochrone 7. Minimum surface of revolution 8. Several dependent variables 9. Parametric representation
    10. Undetermined end points 11. Brachistochrone from a given curve to a fixed point
Chapter 4. Isoperimetric Problems
    1. The simple isoperimetric problem 2. Direct extensions 3. Problem of the maximum enclosed area 4. Shape of a hanging rope. 5. Restrictions imposed through finite or differential equations
Chapter 5. Geometrical Optics: Fermat's Principle
    1. Law of refraction (Snell's law) 2. Fermat's principle and the calculus of variations
Chapter 6. Dynamics of Particles
    1. Potential and kinetic energies. 2. Generalized coordinates 3. Hamilton's principle. Lagrange equations of motion 3. Generalized momenta. Hamilton equations of motion.
    4. Canonical transformations 5. The Hamilton-Jacobi differential equation 6. Principle of least action 7. The extended Hamilton's principle
Chapter 7. Two Independent Variables: The Vibrating String
    1. Extremization of a double integral 2. The vibrating string 3. Eigenvalue-eigenfunction problem for the vibrating string
    4. Eigenfunction expansion of arbitrary functions. Minimum characterization of the eigenvalue-eigenfunction problem 5. General solution of the vibrating-string equation
    6. Approximation of the vibrating-string eigenvalues and eigenfunctions (Ritz method) 7. Remarks on the distinction between imposed and free end-point conditions
Chapter 8. The Sturm-Liouville Eigenvalue-Eigenfunction Problem
    1. Isoperimetric problem leading to a Sturm-Liouville system 2. Transformation of a Sturm-Liouville system 3. Two singular cases: Laguerre polynomials, Bessel functions
Chapter 9. Several Independent Variables: The Vibrating Membrane
    1. Extremization of a multiple integral 2. Change of independent variables. Transformation of the laplacian 3. The vibrating membrane 4. Eigenvalue-eigenfunction problem for the membrane
    5. Membrane with boundary held elastically. The free membrane 6. Orthogonality of the eigenfunctions. Expansion of arbitrary functions 7. General solution of the membrane equation
    8. The rectangular membrane of uniform density 9. The minimum characterization of the membrane eigenvalues 10. Consequences of the minimum characterization of the membrane eigenvalues
    11. The maximum-minimum characterization of the membrane eigenvalues 12. The asymptotic distribution of the membrane eigenvalues 13. Approximation of the membrane eigenvalues
Chapter 10. Theory of Elasticity
    1. Stress and strain 2. General equations of motion and equilibrium 3. General aspects of the approach to certain dynamical problems 4. Bending of a cylindrical bar by couples
    5. Transverse vibrations of a bar 6. The eigenvalue-eigenfunction problem for the vibrating bar 7. Bending of a rectangular plate by couples 8. Transverse vibrations of a thin plate
    9. The eigenvalue-eigenfunction problem for the vibrating plate 10. The rectangular plate. Ritz method of approximation
Chapter 11. Quantum Mechanics
    1. First derivation of the Schrödinger equation for a single particle 2. The wave character of a particle. Second derivation of the Schrödinger equation
    3. The hydrogen atom. Physical interpretation of the Schrödinger wave functions 4. Extension to systems of particles. Minimum character of the energy eigenvalues
    5. Ritz method: Ground state of the helium atom. Hartree model of the many-electron atom
Chapter 12. Electrostatics
    1. Laplace's equation. Capacity of a condenser 2. Approximation of the capacity from below (relaxed boundary conditions) 3. Remarks on problems in two dimensions
    4. The existence of minima of the Dirichlet integral
  Bibliography; Index