Preface |
Chapter 1. |
Introduction |
Chapter 2. |
Background Preliminaries |
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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 |
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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 |
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11. The surface integral 12. Gradient, laplacian 13. Green's theorem (two dimensions) 14. Green's theorem (three dimensions) |
Chapter 3. |
Introductory Problems |
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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 |
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6. The brachistochrone 7. Minimum surface of revolution 8. Several dependent variables 9. Parametric representation |
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10. Undetermined end points 11. Brachistochrone from a given curve to a fixed point |
Chapter 4. |
Isoperimetric Problems |
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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 |
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1. Law of refraction (Snell's law) 2. Fermat's principle and the calculus of variations |
Chapter 6. |
Dynamics of Particles |
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1. Potential and kinetic energies. 2. Generalized coordinates 3. Hamilton's principle. Lagrange equations of motion 3. Generalized momenta. Hamilton equations of motion. |
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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 |
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1. Extremization of a double integral 2. The vibrating string 3. Eigenvalue-eigenfunction problem for the vibrating string |
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4. Eigenfunction expansion of arbitrary functions. Minimum characterization of the eigenvalue-eigenfunction problem 5. General solution of the vibrating-string equation |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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9. The eigenvalue-eigenfunction problem for the vibrating plate 10. The rectangular plate. Ritz method of approximation |
Chapter 11. |
Quantum Mechanics |
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1. First derivation of the Schrödinger equation for a single particle 2. The wave character of a particle. Second derivation of the Schrödinger equation |
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3. The hydrogen atom. Physical interpretation of the Schrödinger wave functions 4. Extension to systems of particles. Minimum character of the energy eigenvalues |
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5. Ritz method: Ground state of the helium atom. Hartree model of the many-electron atom |
Chapter 12. |
Electrostatics |
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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 |
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4. The existence of minima of the Dirichlet integral |
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Bibliography; Index |