| Preface | |
| Introduction | |
| 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 | |
| Determinants | |
| 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 | |
| Geodesics | |
| 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 Schrödinger equation for a single particle | |
| The wave character of a particle. Sec | |
| Table of Contents provided by Publisher. All Rights Reserved. |
