INTRODUCTION TO THE CALCULUS OF VARIATIONS AND ITS APPLICATIONS

Transcription

1 INTRODUCTION TO THE CALCULUS OF VARIATIONS AND ITS APPLICATIONS Frederick Y.M. Wan University of California, Irvine CHAPMAN & HALL I(J)P An International Thomson Publishing Company New York Albany Bonn Boston Cincinnati Detroit London Madrid Melbourne Mexico City Pacific Grove Paris San Francisco Singapore Tokyo Toronto Washington

2 Contents Page Preface xiii 1. The Basic Problem 1. Introduction 1 2. Some Examples 3 3. The Euler Differential Equation 9 4. Integration of the Euler Differential Equation The Brachistochrone Problem Piecewise-Smooth Extremals Exercises Piecewise-Smooth Extremals 1. Piecewise Smooth Solution for the Basic Problem The Euler Lagrange Equation Several Unknowns Parametric Form Erdmann's Corner Conditions The Ultra-Differentiated Form Minimal Surface of Revolution Maximum Rocket Height Exercises Modifications of the Basic Problem 1. The Variational Notation Euler Boundary Conditions Free Boundary Problems 65 VII

3 viii Contents 4. Free and Constrained End Points Higher Derivatives Other End Conditions Exercises A Weak Minimum 1. The Legendre Condition Jacobi's Test Conjugate Points Sufficiency Several Unknowns Convex Integrand Global Minimum Exercises A Strong Minimum 1. A Weak Minimum May Not Be the True Minimum The Weierstrass Excess Function The Figurative Fields of Extremals Sufficiency An Illustrative Example Hilbert's Integral Several Unknowns Exercises 135 Appendix The Hamiltonian 1. The Legendre Transformation and Hamiltonian Systems Hamilton's Principle Canonical Transformations The Hamilton-Jacobi Equation Solutions of the Hamilton-Jacobi Equation The Method of Additive Separation Hamilton's Principal Function Exercises Lagrangian Mechanics 1. Generalized Coordinates Coordinate Transformations 170

4 Contents ix 3. Holonomic Constraints Poisson Brackets Variationally Invariant Lagrangians Noether's Theorem Generators for Variationally Invariant Lagrangians Relativistic Mechanics Exercises Direct Methods 1. The Rayleigh-Ritz Method Completeness and Minimizing Sequence A Weighted Least-Squares Approximation Inhomogeneous End Conditions Piecewise Linear Finite Elements The Finite Element Method Duality The Inverse Problem Weak Solutions Exercises Dynamic Programming 1. The Shortest Route Problem Backward Recursion The Knapsack Problem Forward Recursion Intermediate Knapsack Capacities Vector- and Continuous-State Variables The Variational Problem Exercises Isoperimetric Constraints 1. The Shape of the Hanging Chain Normal Isoperimetric Problems and a Duality Eigenvalue Problems and Mechanical Vibration Variational Formulation of Sturm-Liouville Problems The Rayleigh Quotient Higher Eigenvalues Mixed End Conditions Optimal Harvesting of a Uniform Forest Exercises 289 Appendix 293

5 x Contents 11. Pointwise Constraints on Extremals 1. Pointwise Equality Constraints The Multiplier Rule for Equality Constraints Inequality Constraints on the Unknowns Binding Inequality Constraints Brachistochrone with Limited Descent Inequality Constraints on an End Point Land Use in a Long and Narrow City Exercises Nonholonomic Constraints 1. Equality Constraints Involving Derivatives The Multiplier Rule Brachistochrone in a Resisting Medium Inequality Constraints Singular Solutions The Most Rapid Approach The Hamilton-Jacobi Inequality Blocked Harvest of a Uniform Forest Exercises Optimal Control with Linear Dynamics 1. Optimal Control Statement of the Problem Controllability of Linear Autonomous Systems Nonautonomous Linear Systems Controllability with Constrained Controls An Inventory Control Model A Wheat-Trading Problem The Hamiltonian The Linear Time Optimal Problem Exercises Optimal Control with General Lagrangians 1. The Maximum Principle Controllability of Nonlinear Systems Sustained Consumption with a Finite Resource Deposit The Linear-Quadratic Problem and Feedback Control A Sufficient Condition for Optimality 421

6 Contents xi 6. Household Optimum and Locational Equilibrium The Second Best Residential Land Allocation Perturbation Solution Inequality Constraints Optimality Under Constraints Methods of Jhe Calculus of Variations Exercises 447 Appendix Higher Dimensions 1. The Plateau Problem Euler Differential Equation and Boundary Conditions Sufficient Conditions Dirichlet's Problem on a Unit Disk Several Unknowns Maxwell's Equations Higher Derivatives Finite Elements in Two Dimensions Torsion of Elastic Bars Pointwise Equality Constraints Isoperimetric Constraints Exercises Linear Theory of Elasticity 1. Continuum Mechanics and Elasticity Theory Components of Displacement and Strain Stress Fields and Equilibrium Elasticity and Isotropy Navier's Reduction Minimum Potential Energy Reissner's Variational Principle Minimum Complementary Energy Semi-direct Method Saint-Venant Torsion, Exercises Plate Theory 1. The Elastostatics of Flat Plates The Germain-Kirchhoff Thin Plate Theory The Kirchhoff Contracted Stress Boundary Conditions 539

7 xii Contents 4. A Semi-direct Method of Solution Minimum Complementary Energy Reduction of Reissner's Plate Equations A Variational Principle for Stresses and Displacements Twisting of a Rectangular Plate A Finite Deflection Plate Theory The von Karman Plate Equations Finite Twisting and Bending of Rectangular Plates Exercises Fluid Mechanics Appendix. 1. Mass and Entropy A Lagrangian Variational Principle for Ideal Fluids Ideal Fluid Motion Not Always Irrotational An Eulerian Variational Principle for Ideal Fluids Incompressible Fluids A Surface Wave Problem Slow Dispersion of Wave Trains Creeping Motion of an Incompressible Fluid Oseen's Approximation Exercises 597 Approximate Methods for Euler's Differential Equation 1. Two-Point Boundary-Value Problems Numerical Solution for Initial-Value Problems Linear Boundary-Value Problems The Shooting Method Finite Difference Analogue Accuracy of the Finite Difference Solution Fixed Point Iteration Newton's Iteration Exercises 617 Bibliography 621 Index 627