This text is basically divided into two parts. Chapters 1–4 include background material, basic theorems and isoperimetric problems. Chapters 5–12 are devoted to applications, geometrical optics, particle dynamics, the theory of elasticity, electrostatics, quantum mechanics, and other topics. Exercises in each chapter. 1952 edition.
Author: A. S. GUPTA
Publisher: PHI Learning Pvt. Ltd.
Release Date: 1996-01-01
Calculus of variations is one of the most important mathematical tools of great scientific significance used by scientistis and engineers. Unfortunately, a few books that are available are written at a level which is not easily comprehensible for postgraduate students.This book, written by a highly respected academic, presents the materials in a lucid manner so as to be within the easy grasp of the students with some background in calculus, differential equations and functional analysis. The aim is to give a thorough and systematic analysis of various aspects of calculus of variations.
Applications-oriented introduction to variational theory develops insight and promotes understanding of specialized books and research papers. Suitable for advanced undergraduate and graduate students as a primary or supplementary text. 1969 edition.
Author: Charles Fox
Publisher: Courier Corporation
Release Date: 1987
In this highly regarded text for advanced undergraduate and graduate students, the author develops the calculus of variations both for its intrinsic interest and for its powerful applications to modern mathematical physics. Topics include first and second variations of an integral, generalizations, isoperimetrical problems, least action, special relativity, elasticity, more. 1963 edition.
Author: Charles R. MacCluer
Publisher: Courier Corporation
Release Date: 2013-05-20
First truly up-to-date treatment offers a simple introduction to optimal control, linear-quadratic control design, and more. Broad perspective features numerous exercises, hints, outlines, and appendixes, including a practical discussion of MATLAB. 2005 edition.
This comprehensive text provides all information necessary for an introductory course on the calculus of variations and optimal control theory. Following a thorough discussion of the basic problem, including sufficient conditions for optimality, the theory and techniques are extended to problems with a free end point, a free boundary, auxiliary and inequality constraints, leading to a study of optimal control theory.
Author: Bruce van Brunt
Publisher: Springer Science & Business Media
Release Date: 2006-04-18
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.
This dissertation, "Griffiths' Formalism of the Calculus of Variations and Applications to Invariants" by Hong-Yu, Chow, 周康宇, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of thesis entitled GRIFFITHS' FORMALISM OF THE CALCULUS OF VARIATIONS AND APPLICATIONS TO INVARIANTS submitted by Hong-Yu Chow for the Degree of Master of Philosophy at The University of Hong Kong in August 2005 The calculus of variations is of fundamental importance in many disci- plines including di(R)erential geometry, mechanics, optimization and math- ematical physics. The classical treatment of the subject reduces to the consideration of the system of Euler-Lagrange equations. That is, given a functional J[y] = F, any extremal of J must satisfy the Euler-Lagrange equations, which is a system of di(R)erential equations, provided F satises certain regularity assumptions. The existence of the extremals depends on the solvability of the Euler-Lagrange equations. However, in many practical cases, knowing the existence of extremals is not enough. Finding out the extremal explicitly is even more important. Unfortunately, there are many cases in which the system of Euler-Lagrange equations are dicult, if not impossible, to solve. A simple example is the searching of geodesics on a Riemanniansurface. Inordertosolvesuchtypesofintrinsicgeometricprob- lems, a new formalism via exterior di(R)erential systems was developed by P.Griths. With the help of exterior di(R)erential systems, many intrinsic geo- metric problems of the calculus of variations can be solved e(R)ectively. One exciting application is Cheung's application of the new formalism to search for invariants onC loops. Inthisthesis, theclassicalmethod, togetherwiththeformalismbyGrif- thswerereviewed. Cheung'sapplicationtosearchforinvariantsforthecase of1independentvariableandhisgeneralizationoftheformalismofGriths to the case of several independent variables were also reviewed. By apply- ing the generalization by Cheung, invariants on C parallelizable surfaces are considered. By observing that a functional on a parallizable surface is a C invariant means that the Euler-Lagrange equations associated to the functional are trivial, the di(R)erential topological problem of computing C invariantswastranslatedtotheanalyticproblemofdeterminingwhena functionalwouldhavetrivialEuler-Lagrangeequations. Interestingly, itwas found that, when a compact parallelizable surface is embedded intoR, the only C invariant is the total Gaussian Curvature and nothing else. This resultisanalogoustoCheung'sresult, namely, whenaC loopisembedded intoR, the only invariant is the winding number, that is, the total curva- ture. The method applied in searching for the invariant is an extension to the method of Cheung, making use of the Cheung-Griths formalism for several independent variables. DOI: 10.5353/th_b3581250 Subjects: Calculus of variations
Author: Mike Mesterton-Gibbons
Publisher: American Mathematical Soc.
Release Date: 2009
The calculus of variations is used to find functions that optimize quantities expressed in terms of integrals. Optimal control theory seeks to find functions that minimize cost integrals for systems described by differential equations. This book is an introduction to both the classical theory of the calculus of variations and the more modern developments of optimal control theory from the perspective of an applied mathematician. It focuses on understanding concepts and how to apply them. The range of potential applications is broad: the calculus of variations and optimal control theory have been widely used in numerous ways in biology, criminology, economics, engineering, finance, management science, and physics. Applications described in this book include cancer chemotherapy, navigational control, and renewable resource harvesting. The prerequisites for the book are modest: the standard calculus sequence, a first course on ordinary differential equations, and some facility with the use of mathematical software. It is suitable for an undergraduate or beginning graduate course, or for self study. It provides excellent preparation for more advanced books and courses on the calculus of variations and optimal control theory.
Author: Mariano Giaquinta
Publisher: Springer Science & Business Media
Release Date: 2004-06-23
This two-volume treatise is a standard reference in the field. It pays special attention to the historical aspects and the origins partly in applied problems—such as those of geometric optics—of parts of the theory. It contains an introduction to each chapter, section, and subsection and an overview of the relevant literature in the footnotes and bibliography. It also includes an index of the examples used throughout the book.
This research presents some important domains of partial differential equations and applied mathematics including calculus of variations, control theory, modelling, numerical analysis and various applications in physics, mechanics and engineering. These topics are now part of many areas of science and have experienced tremendous development during the last decades.
Author: John A. Burns
Publisher: CRC Press
Release Date: 2013-08-28
Introduction to the Calculus of Variations and Control with Modern Applications provides the fundamental background required to develop rigorous necessary conditions that are the starting points for theoretical and numerical approaches to modern variational calculus and control problems. The book also presents some classical sufficient conditions and discusses the importance of distinguishing between the necessary and sufficient conditions. In the first part of the text, the author develops the calculus of variations and provides complete proofs of the main results. He explains how the ideas behind the proofs are essential to the development of modern optimization and control theory. Focusing on optimal control problems, the second part shows how optimal control is a natural extension of the classical calculus of variations to more complex problems. By emphasizing the basic ideas and their mathematical development, this book gives you the foundation to use these mathematical tools to then tackle new problems. The text moves from simple to more complex problems, allowing you to see how the fundamental theory can be modified to address more difficult and advanced challenges. This approach helps you understand how to deal with future problems and applications in a realistic work environment.