Calculus of Variations

Suitable for advanced undergraduate and graduate students of mathematics, physics, or engineering, this title presents an introduction to the calculus of variations that focuses on variational problems involving one independent variable. It discusses more advanced topics such as the inverse problem, eigenvalue problems, and Noether's theorem.

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.

From the reviews:

"I find this book a very useful supplementary reading for undergraduate students and a good teaching aid for lecturers of topics involving traditional variational calculus (as e. g. mathematical physics). It is written with a deep pedagogical attention a ] . According to my classroom experience with undergraduate physicists, the presentation of the examples in the book may be very helpful a ] . It can also be appreciated that the author tries to present the results showing motivation and heuristical ideas for each crucial theorem." (L. L. StachA3, Acta Scientiarum Mathematicarum, Vol. 71, 2005)

"The calculus of variations is one of the latest books in Springera (TM)s Universitext series. As such, it is intended to be a non-intimidating, introductory text a ] . I enjoyed reading The calculus of variations. Brunt writes in a lucid, engaging style a ] . can be used in a variety of undergraduate and beginning postgraduate courses. There is sufficient meat, both in the range of examples treated and in the development of the underlying mathematics a ] that most of its intended audience will just be grateful a ] ." (Nick Lord, The Mathematical Gazette, Vol. 89 (516), 2005)

"The author describes this book as suitable for a one semester course for advance undergraduate students in math, physics or engineering. a ] Accordingly, I chose to use this book as my primary reference for presenting the course a ] . From my perspective, the book was pitched at a good level for the students I was teaching a ] . Overall I enjoyed this book, and would unreservedly recommend it a ] . The book really brought home to me the elegance of this subject a ] ." (Matthew Roughan, TheAustralian Mathematical Society Gazette, Vol. 32 (1), 2005)

"This text provides a friendly and a ] elementary introduction to the calculus of variations. a ] The emphasis is on well-chosen examples used to obtain the necessary heuristics for developing the theoretical background. a ] Due to its concrete and well-organized approach, the book constitutes a valuable addition to the text book literature on the calculus of variations." (M. Kunzinger, Monatshefte fA1/4r Mathematik, Vol. 147 (1), 2006)

"Bruce van Brunt shows his love of the subject in his new book The Calculus of Variations a ] . Brunt gives us a nice historical introduction to the calculus of variations. a ] The exercises have a ] been polished and sharpened in the classroom. a ] this is a well crafted, reasonably priced book that would be a fine introduction to a fascinating subject that not enough mathematicians know about." (Ed Sandifer, MathDL, May, 2004)

"Professor van Brunta (TM)s Calculus of Variations is an easily understandable introductory account of the (classical) Calculus of Variations a ] . This text is aimed at the beginning graduate and advanced graduate students of mathematics and physics as well as engineering. a ] The references contain 75 items a ] ." (R. Thiele, Zeitschrift fA1/4r Analysis und ihre Anwendungen, Vol. 24 (4), 2005)

1 Introduction		1	(22)
1.1 Introduction		1	(2)
1.2 The Catenary and Brachystochrone Problems		3	(7)
1.2.1 The Catenary		3	(4)
1.2.2 Brachystochrones		7	(3)
1.3 Hamilton's Principle		10	(4)
1.4 Some Variational Problems from Geometry		14	(7)
1.4.1 Dido's Problem		14	(2)
1.4.2 Geodesics		16	(4)
1.4.3 Minimal Surfaces		20	(1)
1.5 Optimal Harvest Strategy		21	(2)
2 The First Variation		23	(32)
2.1 The Finite-Dimensional Case		23	(5)
2.1.1 Functions of One Variable		23	(3)
2.1.2 Functions of Several Variables		26	(2)
2.2 The Euler-Lagrange Equation		28	(8)
2.3 Some Special Cases		36	(6)
2.3.1 Case I: No Explicit y Dependence		36	(2)
2.3.2 Case II: No Explicit x Dependence		38	(4)
2.4 A Degenerate Case		42	(2)
2.5 Invariance of the Euler-Lagrange Equation		44	(5)
2.6 Existence of Solutions to the Boundary-Value Problem		49	(6)
3 Some Generalizations		55	(18)
3.1 Functionals Containing Higher-Order Derivatives		55	(5)
3.2 Several Dependent Variables		60	(5)
3.3 Two Independent Variables		65	(5)
3.4 The Inverse Problem		70	(3)
4 Isoperimetric Problems		73	(30)
4.1 The Finite-Dimensional Case and Lagrange Multipliers		73	(10)
4.1.1 Single Constraint		73	(4)
4.1.2 Multiple Constraints		77	(2)
4.1.3 Abnormal Problems		79	(4)
4.2 The Isoperimetric Problem		83	(11)
4.3 Some Generalizations on the Isoperimetric Problem		94	(9)
4.3.1 Problems Containing Higher-Order Derivatives		95	(1)
4.3.2 Multiple Isoperimetric Constraints		96	(3)
4.3.3 Several Dependent Variables		99	(4)
5 Applications to Eigenvalue Problems		103	(16)
5.1 The Sturm-Liouville Problem		103	(6)
5.2 The First Eigenvalue		109	(6)
5.3 Higher Eigenvalues		115	(4)
6 Holonomic and Nonholonomic Constraints		119	(16)
6.1 Holonomic Constraints		119	(6)
6.2 Nonholonomic Constraints		125	(6)
6.3 Nonholonomic Constraints in Mechanics		131	(4)
7 Problems with Variable Endpoints		135	(24)
7.1 Natural Boundary Conditions		135	(9)
7.2 The General Case		144	(6)
7.3 Tansversality Conditions		150	(9)
8 The Hamiltonin Formulation		159	(42)
8.1 The Legendre Transformation		160	(4)
8.2 Hamilton's Equations		164	(7)
8.3 Symplectic Maps		171	(4)
8.4 The Hamilton-Jacobi Equation		175	(9)
8.4.1 The General Problem		175	(6)
8.4.2 Conservative Systems		181	(3)
8.5 Separation of Variables		184	(17)
8.5.1 The Method of Additive Separation		185	(5)
8.5.2 Conditions for Separable Solutions		190	(11)
9 Noether's Theorem		201	(20)
9.1 Conservation Laws		201	(1)
9.2 Variational Symmetries		202	(5)
9.3 Noether's Theorem		207	(6)
9.4 Finding Varbational Symmetries		213	(8)
10 The Second Variation		221	(40)
10.1 The Finite-Dimensional Case		221	(3)
10.2 The Second Variation		224	(3)
10.3 The Legendre Condition		227	(5)
10.4 The Jacobi Necessary Condition		232	(9)
10.4.1 A Reformulation of the Second Variation		232	(2)
10.4.2 The Jacobi Accessory Equation		234	(3)
104.3 The Jacobi Necessary Condition		237	(4)
10.5 A Sufficient Condition		241	(3)
10.6 More on Conjugate Points		244	(13)
10.6.1 Finding Conjugate Points		245	(4)
10.6.2 A Geometrical Interpretation		249	(5)
10.6.3 Saddle Points		254	(3)
10.7 Convex Integrands		257	(4)
A Analysis and Differential Equations		261	(12)
A.1 Taylor's Theorem		261	(4)
A.2 The Implicit Function Theorem		265	(3)
A.3 Theory of Ordinary Differential Equations		268	(5)
B Function Spaces		273	(10)
B.1 Normed Spaces		273	(5)
B.2 Banach and Hilbert Spaces		278	(5)
References		283	(4)
Index		287