The calculus of variations变分法的计算
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作者： Bruce van Brunt著
出 版 社：
出版时间： 2003-9-1字数：版次： 1页数： 290印刷时间： 2003/09/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780387402475包装： 精装内容简介
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 familiarity with the elements of real analysis and ordinary differential equations. The book focuses on variational problems that involve one independent variable. The fixed endpoint problem and problems with constraints are discussed in detail. In addition, more advanced topics such as the inverse problem, eigenvalue problems, separability conditions for the Hamilton-Jacobi equation, and Noether's theorem are discussed. The text contains numerous examples to illustrate key concepts along with problems to help the student consolidate the material. The book can be used as a textbook for a one semester course on the calculus of variations, or as a book to supplement a course on applied mathematics or classical mechanics. Bruce van Brunt is Senior Lecturer at Massey University, New Zealand. He is the author of The Lebesgue-Stieltjes Integral, with Michael Carter, and has been teaching the calculus of variations to undergraduate and graduate students for several years.
目录
1 Introduction
1.1 Introduction
1.2 The Catenary and Brachystochrone Problems
1.2.1 The Catenary
1.2.2 Brachystochrones
1.3 Hamilton's Principle
1.4 Some Variational Problems from Geometry
1.4.1 Dido's Problem
1.4.2 Geodesics
1.4.3 Minimal Surfaces
1.5 Optimal Harvest Strategy
2 The First Variation
2.1 The Finite-Dimensional Case
2.1.1 Functions of One Variable
2.1.2 Functions of Several Variables
2.2 The Euler-Lagrange Equation
2.3 Some Special Cases
2.3.1 Case I: No Explicit y Dependence
2.3.2 Case II: No Explicit x Dependence
2.4 A Degenerate Case
2.5 Invariance of the Euler-Lagrange Equation
2.6 Existence of Solutions to the Boundary-Value Problem
3 Some Generalizations
3.1 Functionals Containing Higher-Order Derivatives
3.2 Several Dependent Variables
3.3 Two Independent Variables
3.4 The Inverse Problem
4 Isoperimetric Problems
4.1 The Finite-Dimensional Case and Lagrange Multipliers
4.1.1 Single Constraint
4.1.2 Multiple Constraints
4.1.3 Abnormal Problems
4.2 The Isoperimetric Problem
4.3 Some Generalizations on the Isoperimetric Problem
4.3.1 Problems Containing Higher-Order Derivatives
4.3.2 Multiple Isoperimetric Constraints
4.3.3 Several Dependent Variables
5 Applications to Eigenvalue Problems
5.1 The Sturm-Liouville Problem
5.2 The First Eigenvalue
5.3 Higher Eigenvalues
6 Holonomic and Nonholonomic Constraints
6.1 Holonomic Constraints
6.2 Nonholonomic Constraints
6.3 Nonholonomic Constraints in Mechanics
7 Problems with Variable Endpoints
7.1 Natural Boundary Conditions
7.2 The General Case
7.3 Tansversality Conditions
8 The Hamiltonin Formulation
8.1 The Legendre Transformation
8.2 Hamilton's Equations
8.3 Symplectic Maps
8.4 The Hamilton-Jacobi Equation
8.4.1 The General Problem
8.4.2 Conservative Systems
8.5 Separation of Variables
8.5.1 The Method of Additive Separation
8.5.2 Conditions for Separable Solutions
9 Noether's Theorem
9.1 Conservation Laws
9.2 Variational Symmetries
9.3 Noether's Theorem
9.4 Finding Varbational Symmetries
10 The Second Variation
10.1 The Finite-Dimensional Case
10.2 The Second Variation
10.3 The Legendre Condition
10.4 The Jacobi Necessary Condition
10.4.1 A Reformulation of the Second Variation
10.4.2 The Jacobi Accessory Equation
10.4.3 The Jacobi Necessary Condition
10.5 A Sufficient Condition
10.6 More on Conjugate Points
10.6.1 Finding Conjugate Points
10.6.2 A Geometrical Interpretation
10.6.3 Saddle Points
10.7 Convex Integrands
A Analysis and Differential Equations
A.1 Taylor's Theorem
A.2 The Implicit Function Theorem
A.3 Theory of Ordinary Differential Equations
B Function Spaces
B.1 Normed Spaces
B.2 Banach and Hilbert Spaces
References
Index
