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Calculus of Variations

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ISBN-10: 0387402470

ISBN-13: 9780387402475

Edition: 2004

Authors: Bruce Van Brunt

List price: $99.99
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Description:

The calculus of variations has a long history of interaction with other branches of mathematics, such as geometry and differential equations, and with physics, particularly mechanics. More recently, the calculus of variations has found applications in other fields such as economics and electrical engineering. Much of the mathematics underlying control theory, for instance, can be regarded as part of the calculus of variations.This book is an introductory account of the calculus of variations suitable for advanced undergraduate and graduate students of mathematics, physics, or engineering. The mathematical background assumed of the reader is a course in multivariable calculus, and some…    
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Book details

List price: $99.99
Copyright year: 2004
Publisher: Springer New York
Publication date: 9/12/2003
Binding: Hardcover
Pages: 292
Size: 6.10" wide x 9.25" long x 0.75" tall
Weight: 1.452
Language: English

Introduction
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 Explicity 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
Analysis and Differential Equations
Taylor's Theorem
The Implicit Function Theorem
Theory of Ordinary Differential Equations
Function Spaces
Normed Spaces
Banach and Hilbert Spaces
References
Index