# Introduction To The Calculus of Variations And Its Applications

## 2nd Edition

Routledge

640 pages

Hardback: 9780412051418
pub: 1995-01-01
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eBook (VitalSource) : 9780203749821
pub: 2017-10-19
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### Description

This comprehensive text provides all information necessary for an introductory course on the calculus of variations and optimal control theory. Following a thorough discussion of the basic problem, including sufficient conditions for optimality, the theory and techniques are extended to problems with a free end point, a free boundary, auxiliary and inequality constraints, leading to a study of optimal control theory.

Preface

The Basic Problem

Introduction

Some Examples

The Euler Differential Equation

Integration of the Euler Differential Equation

The Brachistochrone Problem

Pricewise—Smooth Extremals

Exercises

Pricewise—Smooth Extremals

Pricewise—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

The Variational Notation

Euler Boundary Conditions

Free Boundary Problems

Free and Constrained End Points

Higher Derivatives

Other End Conditions

Exercises

A Weak Minimum

The Legendre Condition

Jacobi’s Test

Conjugate Points

Sufficiency

Several Unknowns

Convex Integrand

Global Minimum

Exercises

A Strong Minimum

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

Appendix

The Hamiltonian

The Legendre Transformation and Hamiltonian Systems

Hamilton’s Principle

Canonical Transformations

The Hamiltonian-Jacobi Equation

Solutions of the Hamiltonian-Jacobi Equation

Hamilton’s Principal Function

Exercises

Lagrangian Mechanics

Generalized Coordinates

Coordinate Transformations

Holonomic Constraints

Poisson Brackets

Variationally Invariant Lagrangians

Noeth