Product Details This book by Robert Weinstock was written to fill the need for a basic introduction to the calculus of variations. Simply and easily written, with an emphasis on the applications of this calculus, it has long been a standard reference of physicists, engineers, and applied mathematicians. The author begins slowly, introducing the reader to the calculus of variations, and supplying lists of essential formulae and derivations. Each chapter ends with a series of exercises which should prove very useful in determining whether the material in that chapter has been thoroughly grasped.
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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. Method of undetermined lagrange multipliers 7. The line integral 8. Determinants 9. Formula for surface area The surface integral Gradient, laplacian 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 Undetermined end points 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. Restrictions imposed through finite or differential equations Chapter 5. Dynamics of Particles 1. Potential and kinetic energies.
Generalized coordinates 3. Lagrange equations of motion 3. Generalized momenta. Hamilton equations of motion. Canonical transformations 5. The Hamilton-Jacobi differential equation 6. Principle of least action 7. 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. 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. 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 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 Chapter 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 The rectangular plate.
Ritz method of approximation Chapter Quantum Mechanics 1. The wave character of a particle. The hydrogen atom. 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 Electrostatics 1.
Capacity of a condenser 2. Approximation of the capacity from below relaxed boundary conditions 3. Remarks on problems in two dimensions 4. Back in the United States, Weinstock responded to a call for qualified mathematics instructors at Stanford then, like most American colleges and universities, dealing with a major influx of new students supported by the GI Bill.
He planned at the time to return to academia for only a short time. But, as it turned out, a long teaching career at Stanford, Notre Dame, and finally Oberlin ensued, concluding in after about fifty years. I experienced what were surely the most fascinating eight months of my life. Merchant Marine with a PhD in physics.
Calculus of Variations
Isaac Newton and Gottfried Leibniz also gave some early attention to the subject. An important general work is that of Sarrus which was condensed and improved by Cauchy Other valuable treatises and memoirs have been written by Strauch , Jellett , Otto Hesse , Alfred Clebsch , and Carll , but perhaps the most important work of the century is that of Weierstrass. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation. The 20th and the 23rd Hilbert problem published in encouraged further development.
Calculus Of Variations First Edition
Calculus of variations