## The Calculus Of Variations

Author: Bruce van Brunt
Publisher: Springer Science & Business Media
Release: 2006-04-18
Size: 26.23 MB
Format: PDF
Category : Mathematics
Languages : un
Pages : 292
View: 5560

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.

## A First Course In The Calculus Of Variations

Author: Mark Kot
Publisher: American Mathematical Society
Release: 2014-10-06
Size: 56.43 MB
Format: PDF, Mobi
Category : Mathematics
Languages : un
Pages : 298
View: 4430

This book is intended for a first course in the calculus of variations, at the senior or beginning graduate level. The reader will learn methods for finding functions that maximize or minimize integrals. The text lays out important necessary and sufficient conditions for extrema in historical order, and it illustrates these conditions with numerous worked-out examples from mechanics, optics, geometry, and other fields. The exposition starts with simple integrals containing a single independent variable, a single dependent variable, and a single derivative, subject to weak variations, but steadily moves on to more advanced topics, including multivariate problems, constrained extrema, homogeneous problems, problems with variable endpoints, broken extremals, strong variations, and sufficiency conditions. Numerous line drawings clarify the mathematics. Each chapter ends with recommended readings that introduce the student to the relevant scientific literature and with exercises that consolidate understanding.

## An Introduction To The Calculus Of Variations

Author: Charles Fox
Publisher: Courier Corporation
Release: 1987-01-01
Size: 76.93 MB
Format: PDF, ePub
Category : Mathematics
Languages : un
Pages : 271
View: 5467

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.

## Introduction To The Calculus Of Variations

Author: Bernard Dacorogna
Publisher: Imperial College Press
Release: 2004
Size: 47.67 MB
Format: PDF, ePub, Mobi
Category : Mathematics
Languages : un
Pages : 228
View: 477

- Serves as an excellent introduction to the calculus of variations - Useful to researchers in different fields of mathematics who want to get a concise but broad introduction to the subject - Includes more than 70 exercises with solutions

## Selected Chapters In The Calculus Of Variations

Author: Jürgen Moser
Publisher: Birkhäuser
Release: 2012-12-06
Size: 34.92 MB
Format: PDF, ePub
Category : Mathematics
Languages : un
Pages : 134
View: 4761

0.1 Introduction These lecture notes describe a new development in the calculus of variations which is called Aubry-Mather-Theory. The starting point for the theoretical physicist Aubry was a model for the descrip tion of the motion of electrons in a two-dimensional crystal. Aubry investigated a related discrete variational problem and the corresponding minimal solutions. On the other hand, Mather started with a specific class of area-preserving annulus mappings, the so-called monotone twist maps. These maps appear in mechanics as Poincare maps. Such maps were studied by Birkhoff during the 1920s in several papers. In 1982, Mather succeeded to make essential progress in this field and to prove the existence of a class of closed invariant subsets which are now called Mather sets. His existence theorem is based again on a variational principle. Although these two investigations have different motivations, they are closely re lated and have the same mathematical foundation. We will not follow those ap proaches but will make a connection to classical results of Jacobi, Legendre, Weier strass and others from the 19th century. Therefore in Chapter I, we will put together the results of the classical theory which are the most important for us. The notion of extremal fields will be most relevant. In Chapter II we will investigate variational problems on the 2-dimensional torus. We will look at the corresponding global minimals as well as at the relation be tween minimals and extremal fields. In this way, we will be led to Mather sets.

## A Primer On The Calculus Of Variations And Optimal Control Theory

Author: Mike Mesterton-Gibbons
Publisher: American Mathematical Soc.
Release: 2009
Size: 46.28 MB
Format: PDF, Docs
Category : Mathematics
Languages : un
Pages : 252
View: 7703

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.

## Calculus Of Variations

Author: Robert Weinstock
Publisher: Courier Corporation
Release: 1974-01-01
Size: 16.69 MB
Format: PDF
Category : Mathematics
Languages : un
Pages : 326
View: 100

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.

## A History Of The Calculus Of Variations From The 17th Through The 19th Century

Author: H. H. Goldstine
Publisher: Springer
Release: 2011-10-18
Size: 45.19 MB
Format: PDF, Docs
Category : Mathematics
Languages : un
Pages : 410
View: 3988

The calculus of variations is a subject whose beginning can be precisely dated. It might be said to begin at the moment that Euler coined the name calculus of variations but this is, of course, not the true moment of inception of the subject. It would not have been unreasonable if I had gone back to the set of isoperimetric problems considered by Greek mathemati cians such as Zenodorus (c. 200 B. C. ) and preserved by Pappus (c. 300 A. D. ). I have not done this since these problems were solved by geometric means. Instead I have arbitrarily chosen to begin with Fermat's elegant principle of least time. He used this principle in 1662 to show how a light ray was refracted at the interface between two optical media of different densities. This analysis of Fermat seems to me especially appropriate as a starting point: He used the methods of the calculus to minimize the time of passage cif a light ray through the two media, and his method was adapted by John Bernoulli to solve the brachystochrone problem. There have been several other histories of the subject, but they are now hopelessly archaic. One by Robert Woodhouse appeared in 1810 and another by Isaac Todhunter in 1861.

## The Inverse Problem Of The Calculus Of Variations

Author: Dmitry V. Zenkov
Publisher: Atlantis Press
Release: 2015-10-27
Size: 32.85 MB
Format: PDF, ePub, Mobi
Category : Mathematics
Languages : un
Pages : 289
View: 495

The aim of the present book is to give a systematic treatment of the inverse problem of the calculus of variations, i.e. how to recognize whether a system of differential equations can be treated as a system for extremals of a variational functional (the Euler-Lagrange equations), using contemporary geometric methods. Selected applications in geometry, physics, optimal control, and general relativity are also considered. The book includes the following chapters: - Helmholtz conditions and the method of controlled Lagrangians (Bloch, Krupka, Zenkov) - The Sonin-Douglas's problem (Krupka) - Inverse variational problem and symmetry in action: The Ostrogradskyj relativistic third order dynamics (Matsyuk.) - Source forms and their variational completion (Voicu) - First-order variational sequences and the inverse problem of the calculus of variations (Urban, Volna) - The inverse problem of the calculus of variations on Grassmann fibrations (Urban).

## Calculus Of Variations

Author: Izrail Moiseevitch Gelfand
Publisher: Courier Corporation
Release: 2000-01-01
Size: 72.19 MB
Format: PDF, ePub
Category : Mathematics
Languages : un
Pages : 232
View: 5695

Fresh, lively text serves as a modern introduction to the subject, with applications to the mechanics of systems with a finite number of degrees of freedom.Ideal for math and physics students.