User:Julius/Sandbox

Subject Matter

 * Calculus of Variations

Contents

 * I. Introduction


 * II. The First Variation


 * III. Some Generalizations


 * IV. Isoperimetric Problems


 * V. Applications to Eigenvalue Problems


 * VI. Holonomic and Nonholonomic Constraints


 * VII. Problems with Variable Endpoints


 * VIII. The Hamiltonian Formulation


 * IX. Noether's Theorem


 * X. The Second Variation


 * A. Analysis and Differential Equations


 * B. Function Spaces


 * References


 * Index

Subject Matter

 * Calculus of Variations

Contents
Preface

Acknowledgments


 * I. Calculus of Variations


 * 1. Historical Notes on the Calculus of Variations


 * 2. Introduction and Preliminaries


 * 3. The Simplest Problem in the Calculus of Variations


 * 4. Necessary Conditions for Local Minima


 * 5. Sufficient Conditions for the Simplest Problem


 * 6. Summary for the Simplest Problem


 * 7. Extensions and Generalizations


 * 8. Applications


 * II. Optimal Control


 * 9. Optimal Control Problems


 * 10. Simplest Problem in Optimal Control


 * 11. Extensions of the Maximum Principle


 * 12. Linear Control Systems

Bibliography

Index

Subject Matter

 * Calculus of Variations

Contents
Preface

Acknowledgments


 * I. Calculus of Variations


 * 1. Historical Notes on the Calculus of Variations


 * 2. Introduction and Preliminaries


 * 3. The Simplest Problem in the Calculus of Variations


 * 4. Necessary Conditions for Local Minima


 * 5. Sufficient Conditions for the Simplest Problem


 * 6. Summary for the Simplest Problem


 * 7. Extensions and Generalizations


 * 8. Applications


 * II. Optimal Control


 * 9. Optimal Control Problems


 * 10. Simplest Problem in Optimal Control


 * 11. Extensions of the Maximum Principle


 * 12. Linear Control Systems

Bibliography

Index

Subject Matter

 * Calculus of Variations

Contents
Symbols and Notation

PART 1. CALCULUS OF VARITAIONS

1. Introduction

2. Problem Statement and Necessary Conditions for an Extremum

3. Integration of the Euler - Lagrange Equation

4. An Inverse Problem

5. The Weierstrass Necessary Condition

6. Jacobi's Necessary Condition

7. Corner Conditions

8. Concluding Remarks

PART II. OPTIMAL CONTROL

9. Introduction

10. Problem Statement and Optimality

11. Regular Optimal Trajectories

12. Examples of Extremal Control

13. Some Generalizations

14. Special Systems

15. Sufficient Conditions

16. Feedback Control

17. Optimization with Vector - Valued Cost

REFERENCES

BIBLIOGRAPHY

INDEX

Subject Matter

 * Calculus of Variations

Contents
Part I: Measure Theory and $ L^p $ Spaces

1. Measures

2. $ L^p $ Spaces

Part II: The Direct Method and Lower Semicontinuity

3. The Direct Method and Lower Semicontinuity

4. Convex Analysis

Part III: Functional Defined on $ L^p $

5. Integrands $ f = f \left ( { z } \right ) $

6. Integrands $ f = f \left ( { x, z } \right ) $

7. Integrands $ f = f \left ( { x, u, z } \right ) $

8. Young Measures

Part IV: Appendix

A. Functional Analysis and Set Theory

B. Notes and Open Problems

Notations and List of Symbols

Acknowledgments

References

Index

Subject Matter

 * Calculus of Variations

Contents
Preface

Chapter 1. Introduction

Chapter 2. The First Variation

Chapter 3. Cases and Examples

Chapter 4. Basic Generalizations

Chapter 5. Constraints

Chapter 6. The Second Variation

Chapter 7. Review and Preview

Chapter 8. The Homogenous Problem

Chapter 9. Variable-Endpoint Conditions

Chapter 10. Broken Extremals

Chapter 11. Strong Variations

Chapter 12. Sufficient Conditions

Bibliography

Index

Subject Matter

 * Calculus of Variations

Contents
Part I. Simpler Problems of the Calculus of Variations

I. The Calculus of Variations of Three-Space

II. Sufficient Conditions for a Minimum

III. Fields and the Hamilton-Jacobi Theory

IV. Problems in the Plane and in Higher Spaces

V. Problems in Parametric Form

VI. Problems with Variable End-Points

Part II. The Problem of Bolza

VIII. Further Necessary Conditions for a Minimum

IX. Sufficient Conditions for a Minimum

Appendix

Appendix. Existence Theorems for Implicit Functions and Differential Equations

I. Existence Theorems for Implicit Functions

II. Existence Theorems for Differential Equations

Bibliography

A Bibliography for the Problem of Bolza

Index

Subject Matter

 * Calculus of Variations

Contents
INTRODUCTION

CHAPTER I. INTEGRALS OF THE FIRST ORDER; MAXIMA AND MINIMA FOR SPECIAL WEAK VARIATIONS; EULER TEST, LEGENDRE TEST, JACOBI TEST.

CHAPTER II. INTEGRALS OF THE FIRST ORDER; GENERAL WEAK VARIATIONS; THE METHOD OF WEIERSTRASS

CHAPTER III. INTEGRALS INVOLVING DERIVATIVES OF THE SECOND ORDER; SPECIAL WEAK VARIATIONS, BY THE METHOD OF JACOBI; GENERAL WEAK VARIATIONS, BY THE METHOD OF WEIERSTRASS

CHAPTER IV. INTEGRALS INVOLVING TWO DEPENDENT VARIABLES AND THEIR FIRST DERIVATIVES; SPECIAL WEAK VARIATIONS.

CHAPTER V. INTEGRALS INVOLVING TWO DEPENDENT VARIABLES AND THEIR FIRST DERIVATIVES; GENERAL WEAK VARIATIONS.

CHAPTER VI. INTEGRALS WITH TWO DEPENDENT VARIABLES AND DERIVATIVES OF THE SECOND ORDER; MAINLY SPECIAL WEAK VARIATIONS.

CHAPTER VII. ORDINARY INTEGRALS UNDER STRONG VARIATIONS, AND THE WEIERSTRASS TEST; SOLID OF LEAST RESISTANCE; ACTION

CHAPTER VIII. RELATIVE MAXIMA AND MINIMA OF SINGLE INTEGRALS; ISOPERIMETRICAL PROBLEMS

CHAPTER IX. DOUBLE INTEGRALS WITH DERIVATIVES OF THE FIRST ORDER; WEAK VARIATIONS; MINIMAL SURFACES

CHAPTER X. STRONG VARIATIONS AND THE WEIERSTRASS TEST, FOR DOUBLE INTEGRALS INVOLVING FIRST DERIVATIVES; ISOPERIMETRICAL PROBLEMS

CHAPTER XI. DOUBLE INTEGRALS, WITH DERIVATIVES OF THE SECOND ORDER; WEAK VARIATIONS

CHAPTER XII. TRIPLE INTEGRALS WITH FIRST DERIVATIVES

INDEX

Subject Matter

 * Calculus of Variations

Contents
Part I. The First Variation and Necessary Conditions

Chapter 1. The First Variation

Chapter 2. Variational Problems with Subsidiary Conditions

Chapter 3. General Variational Formulas

Part II. The Second Variation and Sufficient Conditions

Chapter 4. Second Variation, Excess Function, Convexity

Chapter 5. Weak Minimizers and Jacobi Theory

Chapter 6. Weierstrass Field Theory for One-Dimensional Integrals and Strong Minimizers

Supplement. Some Facts from Differential Geometry and Analysis

A List of Examples

Bibliography

Subject Index

Subject Matter

 * Calculus of Variations

Contents
Part III. Canonical Formalism and Parametric Variational Problems

Chapter 7. Legendre Transformation, Hamiltonian Systems, Convexity, Field Theory

Chapter 8. Parametric Variational Integrals

Part IV. Hamilton-Jacobi Theory and Partial Differential Equations of First Order

Chapter 9. Hamilton-Jacobi Theory and Canonical Transformations

Chapter 10. Partial Differential Equations of First Order and Contact Transformations

A List of Examples

A Glimpse at the Literature

Bibliography

Subject Index