The Calculus of Variations - Universitext Softcover Reprint of the Original 1st Ed. 2004 edition

Description for Sales People: 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. Review Quotes: 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 . According to my classroom experience with undergraduate physicists, the presentation of the examples in the book may be very helpful . It can also be appreciated that the author tries to present the results showing motivation and heuristical ideas for each crucial theorem." (L. L. Stacho, Acta Scientiarum Mathematicarum, Vol. 71, 2005) "The calculus of variations is one of the latest books in Springer s Universitext series. As such, it is intended to be a non-intimidating, introductory text . I enjoyed reading The calculus of variations. Brunt writes in a lucid, engaging style . 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 that most of its intended audience will just be grateful ." (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. Accordingly, I chose to use this book as my primary reference for presenting the course . From my perspective, the book was pitched at a good level for the students I was teaching . Overall I enjoyed this book, and would unreservedly recommend it . The book really brought home to me the elegance of this subject ." (Matthew Roughan, The Australian Mathematical Society Gazette, Vol. 32 (1), 2005) "This text provides a friendly and elementary introduction to the calculus of variations. The emphasis is on well-chosen examples used to obtain the necessary heuristics for developing the theoretical background. Due to its concrete and well-organized approach, the book constitutes a valuabReview Quotes: 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 . According to my classroom experience with undergraduate physicists, the presentation of the examples in the book may be very helpful . It can also be appreciated that the author tries to present the results showing motivation and heuristical ideas for each crucial theorem." (L. L. Stacho, Acta Scientiarum Mathematicarum, Vol. 71, 2005) "The calculus of variations is one of the latest books in Springer s Universitext series. As such, it is intended to be a non-intimidating, introductory text . I enjoyed reading The calculus of variations. Brunt writes in a lucid, engaging style . 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 that most of its intended audience will just be grateful ." (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. Accordingly, I chose to use this book as my primary reference for presenting the course . From my perspective, the book was pitched at a good level for the students I was teaching . Overall I enjoyed this book, and would unreservedly recommend it . The book really brought home to me the elegance of this subject ." (Matthew Roughan, The Australian Mathematical Society Gazette, Vol. 32 (1), 2005) "This text provides a friendly and elementary introduction to the calculus of variations. The emphasis is on well-chosen examples used to obtain the necessary heuristics for developing the theoretical background. Due to its concrete and well-organized approach, the book cTable of Contents: 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 Explicity 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. Transversality Conditions -- 8. The Hamiltonian 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 Variational 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. Publisher Marketing: Suitable for advanced undergraduate and graduate students of mathematics, physics, or engineering, this introduction to the calculus of variations focuses on variational problems involving one independent variable. It also discusses more advanced topics such as the inverse problem, eigenvalue problems, and Noether's theorem. The text includes numerous examples along with problems to help students consolidate the material.