Calculus of Variations has 19 ratings and 0 reviews. This text is basically divided into two parts. Chapters 1–4 include background material. Calculus of Variations, With Applications to Physics and Engineering – Weinstock R – Free ebook download as PDF File .pdf) or read book online for free. Synge, J. L. Review: Robert Weinstock, Calculus of variations with applications to physics and engineering. Bull. Amer. Math. Soc. 59 (), no. 4,
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Calculus of Variations by Robert Weinstock
The author begins slowly, introducing the reader to the calculus of variations, and supplying lists of essential formulae and derivations.
Later chapters cover isoperimetric problems, geometrical optics, Fermat’s principle, dynamics of particles, the Sturm-Liouville eigenvalue-eigenfunction problem, the theory of elasticity, quantum mechanics, and electrostatics.
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. The clarity of exposition makes this book easily accessible to anyone who has mastered first-year calculus with some exposure to ordinary differential equations. Physicists and engineers who find variational methods evasive at times will find this book particularly helpful. Synge, Bulletin of the American Mathematical Society.
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Table of contents Preface Chapter 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. Formula for surface area Taylor’s theorem for functions of several variables The surface integral Green’s theorem two dimensions Green’s theorem three dimensions Chapter 3. A basic lemma 2. Statement and formulation of several problems 3. The Varlations equation 4. First integrals of the Euler-Lagrange equation. A degenerate case 5.
Calculus Of Variations First Edition
Minimum surface of revolution 8. Several dependent variables 9. Undetermined end points Brachistochrone from a given curve to a fixed point Chapter 4.
The simple isoperimetric problem 2. Problem of the maximum enclosed area 4. Shape of a hanging rope. Restrictions imposed through finite or differential equations Chapter 5.
Law of refraction Snell’s variafions 2. Fermat’s principle and the calculus of variations Chapter 6. Dynamics of Particles 1. Potential and kinetic energies.
Lagrange equations of motion 3. Hamilton equations of motion. The Hamilton-Jacobi differential equation 6. Principle of least action 7. The extended Hamilton’s principle Chapter 7. 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.
Isoperimetric problem leading to a Sturm-Liouville system 2. Transformation of a Sturm-Liouville system 3.
Laguerre polynomials, Bessel functions Chapter 9. The Vibrating Membrane 1. Extremization of a multiple integral 2. Change of independent variables. Transformation of the laplacian 3. The vibrating membrane 4.
Calculus of Variations : Robert Weinstock :
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.