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Preface:

Introduction

The Catenary and Brachystochrone Problems

The Catenary

Brachystochrones

Hamilton’s Principle

Some Variational Problems from Geometry

Dido’s Problem

Geodesics

Minimal Surfaces

Optimal Harvest Strategy

The First Variation

The Finite-Dimensional Case

Functions of One Variable

Functions of Several Variables

The Euler-Lagrange Equation

Some Special Cases

Case I: No Explicit y Dependence

Case II: No Explicit x Dependence

A Degenerate Case

Invariance of the Euler-Lagrange Equation

Existence of Solutions to the Boundary-Value Problem

Some Generalizations

Functionals Containing Higher-Order Derivatives

Several Dependent Variables

Two Independent Variables

The Inverse Problem

Isoperimetric Problems

The Finite-Dimensional Case and Lagrange Multipliers

Single Constraint

Multiple Constraints

Abnormal Problems

The Isoperimetric Problem

Some Generalizations on the Isoperimetric Problem

Problems Containing Higher-Order Derivatives

Multiple Isoperimetric Constraints

Several Dependent Variables

Applications to Eigenvalue Problems

The Sturm-Liouville Problem

The First Eigenvalue

Higher Eigenvalues

Holonomic and Nonholonomic Constraints

Holonomic Constraints

Nonholonomic Constraints

Nonholonomic Constraints in Mechanics

Problems with Variable Endpoints

Natural Boundary Conditions

The General Case

Transversality Conditions

The Hamiltonian Formulation

The Legendre Transformation

Hamilton’s Equations

Symplectic Maps

The Hamilton-Jacobi Equation

The General Problem

Conservative Systems

Separation of Variables

The Method of Additive Separation

Conditions for Separable Solutions

Noether’s Theorem

Conservation Laws

Variational Symmetries

Noether’s Theorem

Finding Variational Symmetries

The Second Variation

The Finite-Dimensional Case

The Second Variation

The Legendre Condition

The Jacobi Necessary Condition

A Reformulation of the Second Variation

The Jacobi Accessory Equation

The Jacobi Necessary Condition

A Sufficient Condition

More on Conjugate Points

Finding Conjugate Points

A Geometrical Interpretation

Saddle Points

Convex Integrands

A Analysis and Differential Equations

ATaylor’s Theorem

AThe Implicit Function Theorem

ATheory of Ordinary Differential Equations

B Function Spaces

BNormed Spaces

BBanach and Hilbert Spaces