Introduction to the Modern Calculus of Variations
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Cloaking Via Change of Variables for Second Order Quasi-Linear Elliptic Differential Equations Maher Belgacem, Abderrahman Boukricha
Cloaking via change of variables for second order quasi-linear elliptic differential equations Maher Belgacem, Abderrahman Boukricha To cite this version: Maher Belgacem, Abderrahman Boukricha. Cloaking via change of variables for second order quasi- linear elliptic differential equations. 2017. hal-01444772 HAL Id: hal-01444772 https://hal.archives-ouvertes.fr/hal-01444772 Preprint submitted on 24 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Cloaking via change of variables for second order quasi-linear elliptic differential equations Maher Belgacem, Abderrahman Boukricha University of Tunis El Manar, Faculty of Sciences of Tunis 2092 Tunis, Tunisia. Abstract The present paper introduces and treats the cloaking via change of variables in the framework of quasi-linear elliptic partial differential operators belonging to the class of Leray-Lions (cf. [14]). We show how a regular near-cloak can be obtained using an admissible (nonsingular) change of variables and we prove that the singular change- of variable-based scheme achieve perfect cloaking in any dimension d ≥ 2. We thus generalize previous results (cf. [7], [11]) obtained in the context of electric impedance tomography formulated by a linear differential operator in divergence form. -
CALCULUS of VARIATIONS and TENSOR CALCULUS
CALCULUS OF VARIATIONS and TENSOR CALCULUS U. H. Gerlach September 22, 2019 Beta Edition 2 Contents 1 FUNDAMENTAL IDEAS 5 1.1 Multivariable Calculus as a Prelude to the Calculus of Variations. 5 1.2 Some Typical Problems in the Calculus of Variations. ...... 6 1.3 Methods for Solving Problems in Calculus of Variations. ....... 10 1.3.1 MethodofFiniteDifferences. 10 1.4 TheMethodofVariations. 13 1.4.1 Variants and Variations . 14 1.4.2 The Euler-Lagrange Equation . 17 1.4.3 Variational Derivative . 20 1.4.4 Euler’s Differential Equation . 21 1.5 SolvedExample.............................. 24 1.6 Integration of Euler’s Differential Equation. ...... 25 2 GENERALIZATIONS 33 2.1 Functional with Several Unknown Functions . 33 2.2 Extremum Problem with Side Conditions. 38 2.2.1 HeuristicSolution. 40 2.2.2 Solution via Constraint Manifold . 42 2.2.3 Variational Problems with Finite Constraints . 54 2.3 Variable End Point Problem . 55 2.3.1 Extremum Principle at a Moment of Time Symmetry . 57 2.4 Generic Variable Endpoint Problem . 60 2.4.1 General Variations in the Functional . 62 2.4.2 Transversality Conditions . 64 2.4.3 Junction Conditions . 66 2.5 ManyDegreesofFreedom . 68 2.6 Parametrization Invariant Problem . 70 2.6.1 Parametrization Invariance via Homogeneous Function .... 71 2.7 Variational Principle for a Geodesic . 72 2.8 EquationofGeodesicMotion . 76 2.9 Geodesics: TheirParametrization.. 77 3 4 CONTENTS 2.9.1 Parametrization Invariance. 77 2.9.2 Parametrization in Terms of Curve Length . 78 2.10 Physical Significance of the Equation for a Geodesic . ....... 80 2.10.1 Freefloatframe ........................ -
Leibniz's Differential Calculus Applied to the Catenary
Leibniz’s differential calculus applied to the catenary Olivier Keller, agrégé in mathematics, PhD (EHESS) Study of the catenary was a response to a challenge laid down by Jacques Bernoulli, and which was successfully met by Leibniz as well as by Jean Bernoulli and Huygens: to find the curve described by a piece of strung suspended from its two ends. Stimulated by the success of this initial research, Jean Bernoulli put forward and solved similar problems: the form taken by a horizontal blade immobilised on one side, with a weight attached to the other; the form taken by a piece of linen filled with liqueur; the curve of a sail. Challenges among scholars In the 17th century, it was customary for scholars to set each other challenges in alternate issues of journals. Leibniz, for example, challenged the Cartesians – as part of the controversy surrounding the laws of collision, known as the “querelle des forces vives” (vis viva controversy) – to find the curve down which a body falls at constant vertical velocity (isochrone curve). Put forward in the Nouvelle République des Lettres of September 1687, this problem received Huygens’ solution in October of the same year, while Leibniz’s appeared in 1689 in the journal he had created, the Acta Eruditorum. Another famous example is that of the brachistochrone, or the “curve of the fastest descent”, which, thanks to the new differential calculus, proved to be not a circle, as Galileo had believed, but a cycloid: the challenge had been set by Jean Bernoulli in the Acta Eruditorum in June 1696, and was resolved by Leibniz in May 1697. -
José M. Vega-Guzmán
Jos´eM. Vega-Guzm´an Curriculum Vitae Contact Department of Mathematics Tel: (409) 880-8792 Information College of Arts and Sciences Fax: (409) 880-8794 Lamar University email: [email protected] 200-C Lucas Building URL: https://sites.google.com/site/profjmvg/ Beaumont, TX 77710 Education PhD (Applied Mathematics) 2008{2013 • Arizona State University, Tempe, AZ • Co-Advisors: Sergei K. Suslov and Carlos Castillo-Chavez • Dissertation Title: Solution Methods for Certain Evolution Equations Master in Natural Sciences (Mathematics) 2006{2008 • Arizona State University, Tempe, AZ B.S. (MathEd) 2000{2005 • Universidad de Puerto Rico, Cayey, PR • Advisor: Maria A. Avi~n´o Research Interests Nonlinear Wave Propagation. Quantum Mechanics. Integrability of Nonlinear Evolution Equations. Applications of ODE's and PDE's in the Natural Sciences. Mathematical Modeling. Publications 2017 Vega{Guzm´an,J.M.; Alqahtani, R.; Zhou, Q.; Mahmood, M.F.; Moshokoa, S.P.; Ullah, M.Z.; Biswas, A.; Belic, M. Optical solitons for Lakshmanan-Porsezian-Daniel model with spatio-temporal dispersion using the method of undetermined coefficients. Optik - International Journal for Light and Electron Optics. 144, 115{123. 2017 Wang, G.; Kara, A.H.; Vega{Guzm´an,J.M.; Biswas, A. Group analysis, nonlinear self-adjointness, conser- vation laws and soliton solutions for the mKdV systems. Nonlinear Analysis: Modeling and Control. 22(3), 334{346. 2017 Vega{Guzm´an,J.M.; Ullah, M.Z.; Asma, M.; Zhou, Q.; Biswas, A. Dispersive solitons in magneto-optic waveguides. Superlattices and Microstructures. 103, 161{170. 2017 Vega{Guzm´an,J.M.; Mahmood, M.F.; Zhou, Q.; Triki, H.; Arnous, A.H.; Biswas, A.; Moshokoa, S.P.; Belic, M. -
Chapter 8 Change of Variables, Parametrizations, Surface Integrals
Chapter 8 Change of Variables, Parametrizations, Surface Integrals x0. The transformation formula In evaluating any integral, if the integral depends on an auxiliary function of the variables involved, it is often a good idea to change variables and try to simplify the integral. The formula which allows one to pass from the original integral to the new one is called the transformation formula (or change of variables formula). It should be noted that certain conditions need to be met before one can achieve this, and we begin by reviewing the one variable situation. Let D be an open interval, say (a; b); in R , and let ' : D! R be a 1-1 , C1 mapping (function) such that '0 6= 0 on D: Put D¤ = '(D): By the hypothesis on '; it's either increasing or decreasing everywhere on D: In the former case D¤ = ('(a);'(b)); and in the latter case, D¤ = ('(b);'(a)): Now suppose we have to evaluate the integral Zb I = f('(u))'0(u) du; a for a nice function f: Now put x = '(u); so that dx = '0(u) du: This change of variable allows us to express the integral as Z'(b) Z I = f(x) dx = sgn('0) f(x) dx; '(a) D¤ where sgn('0) denotes the sign of '0 on D: We then get the transformation formula Z Z f('(u))j'0(u)j du = f(x) dx D D¤ This generalizes to higher dimensions as follows: Theorem Let D be a bounded open set in Rn;' : D! Rn a C1, 1-1 mapping whose Jacobian determinant det(D') is everywhere non-vanishing on D; D¤ = '(D); and f an integrable function on D¤: Then we have the transformation formula Z Z Z Z ¢ ¢ ¢ f('(u))j det D'(u)j du1::: dun = ¢ ¢ ¢ f(x) dx1::: dxn: D D¤ 1 Of course, when n = 1; det D'(u) is simply '0(u); and we recover the old formula. -
Digital Adaptions of the Scores for Cage Variations I, II and III
Edith Cowan University Research Online ECU Publications 2012 1-1-2012 Digital adaptions of the scores for Cage Variations I, II and III Lindsay Vickery Catherine Hope Edith Cowan University Stuart James Edith Cowan University Follow this and additional works at: https://ro.ecu.edu.au/ecuworks2012 Part of the Music Commons Vickery, L. R., Hope, C. A., & James, S. G. (2012). Digital adaptions of the scores for Cage Variations I, II and III. Proceedings of International Computer Music Conference. (pp. 426-432). Ljubljana. International Computer Music Association. Available here This Conference Proceeding is posted at Research Online. https://ro.ecu.edu.au/ecuworks2012/165 NON-COCHLEAR SOUND _ I[M[2012 LJUBLJANA _9.-14. SEP'l'EMBER Digital adaptions of the scores for Cage Variations I, II and III Lindsay Vickery, Cat Hope, and Stuart James Western Australian Academy of Performing Arts, Edith Cowan University ABSTRACT Over the ten years from 1958 to 1967, Cage revisited to the Variations series as a means of expanding his Western Australian new music ensemble Decibel have investigation not only of nonlinear interaction with the devised a software-based tool for creating realisations score but also of instrumentation, sonic materials, the of the score for John Cage's Variations I and II. In these performance space and the environment The works works Cage had used multiple transparent plastic sheets chart an evolution from the "personal" sound-world of with various forms of graphical notation, that were the performer and the score, to a vision potentially capable of independent positioning in respect to one embracing the totality of sound on a global scale. -
ON the BRACHISTOCHRONE PROBLEM 1. Introduction. The
ON THE BRACHISTOCHRONE PROBLEM OLEG ZUBELEVICH STEKLOV MATHEMATICAL INSTITUTE OF RUSSIAN ACADEMY OF SCIENCES DEPT. OF THEORETICAL MECHANICS, MECHANICS AND MATHEMATICS FACULTY, M. V. LOMONOSOV MOSCOW STATE UNIVERSITY RUSSIA, 119899, MOSCOW, MGU [email protected] Abstract. In this article we consider different generalizations of the Brachistochrone Problem in the context of fundamental con- cepts of classical mechanics. The correct statement for the Brachis- tochrone problem for nonholonomic systems is proposed. It is shown that the Brachistochrone problem is closely related to vako- nomic mechanics. 1. Introduction. The Statement of the Problem The article is organized as follows. Section 3 is independent on other text and contains an auxiliary ma- terial with precise definitions and proofs. This section can be dropped by a reader versed in the Calculus of Variations. Other part of the text is less formal and based on Section 3. The Brachistochrone Problem is one of the classical variational prob- lems that we inherited form the past centuries. This problem was stated by Johann Bernoulli in 1696 and solved almost simultaneously by him and by Christiaan Huygens and Gottfried Wilhelm Leibniz. Since that time the problem was discussed in different aspects nu- merous times. We do not even try to concern this long and celebrated history. 2000 Mathematics Subject Classification. 70G75, 70F25,70F20 , 70H30 ,70H03. Key words and phrases. Brachistochrone, vakonomic mechanics, holonomic sys- tems, nonholonomic systems, Hamilton principle. The research was funded by a grant from the Russian Science Foundation (Project No. 19-71-30012). 1 2 OLEG ZUBELEVICH This article is devoted to comprehension of the Brachistochrone Problem in terms of the modern Lagrangian formalism and to the gen- eralizations which such a comprehension involves. -
On the Ekeland Variational Principle with Applications and Detours
Lectures on The Ekeland Variational Principle with Applications and Detours By D. G. De Figueiredo Tata Institute of Fundamental Research, Bombay 1989 Author D. G. De Figueiredo Departmento de Mathematica Universidade de Brasilia 70.910 – Brasilia-DF BRAZIL c Tata Institute of Fundamental Research, 1989 ISBN 3-540- 51179-2-Springer-Verlag, Berlin, Heidelberg. New York. Tokyo ISBN 0-387- 51179-2-Springer-Verlag, New York. Heidelberg. Berlin. Tokyo No part of this book may be reproduced in any form by print, microfilm or any other means with- out written permission from the Tata Institute of Fundamental Research, Colaba, Bombay 400 005 Printed by INSDOC Regional Centre, Indian Institute of Science Campus, Bangalore 560012 and published by H. Goetze, Springer-Verlag, Heidelberg, West Germany PRINTED IN INDIA Preface Since its appearance in 1972 the variational principle of Ekeland has found many applications in different fields in Analysis. The best refer- ences for those are by Ekeland himself: his survey article [23] and his book with J.-P. Aubin [2]. Not all material presented here appears in those places. Some are scattered around and there lies my motivation in writing these notes. Since they are intended to students I included a lot of related material. Those are the detours. A chapter on Nemyt- skii mappings may sound strange. However I believe it is useful, since their properties so often used are seldom proved. We always say to the students: go and look in Krasnoselskii or Vainberg! I think some of the proofs presented here are more straightforward. There are two chapters on applications to PDE. -
Finite Elements for the Treatment of the Inextensibility Constraint
Comput Mech DOI 10.1007/s00466-017-1437-9 ORIGINAL PAPER Fiber-reinforced materials: finite elements for the treatment of the inextensibility constraint Ferdinando Auricchio1 · Giulia Scalet1 · Peter Wriggers2 Received: 5 April 2017 / Accepted: 29 May 2017 © Springer-Verlag GmbH Germany 2017 Abstract The present paper proposes a numerical frame- 1 Introduction work for the analysis of problems involving fiber-reinforced anisotropic materials. Specifically, isotropic linear elastic The use of fiber-reinforcements for the development of novel solids, reinforced by a single family of inextensible fibers, and efficient materials can be traced back to the 1960s are considered. The kinematic constraint equation of inex- and, to date, it is an active research topic. Such materi- tensibility in the fiber direction leads to the presence of als are composed of a matrix, reinforced by one or more an undetermined fiber stress in the constitutive equations. families of fibers which are systematically arranged in the To avoid locking-phenomena in the numerical solution due matrix itself. The interest towards fiber-reinforced mate- to the presence of the constraint, mixed finite elements rials stems from the mechanical properties of the fibers, based on the Lagrange multiplier, perturbed Lagrangian, and which offer a significant increase in structural efficiency. penalty method are proposed. Several boundary-value prob- Particularly, these materials possess high resistance and lems under plane strain conditions are solved and numerical stiffness, despite the low specific weight, together with an results are compared to analytical solutions, whenever the anisotropic mechanical behavior determined by the fiber derivation is possible. The performed simulations allow to distribution. -
7. Transformations of Variables
Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 7. Transformations of Variables Basic Theory The Problem As usual, we start with a random experiment with probability measure ℙ on an underlying sample space. Suppose that we have a random variable X for the experiment, taking values in S, and a function r:S→T . Then Y=r(X) is a new random variabl e taking values in T . If the distribution of X is known, how do we fin d the distribution of Y ? This is a very basic and important question, and in a superficial sense, the solution is easy. 1. Show that ℙ(Y∈B) = ℙ ( X∈r −1( B)) for B⊆T. However, frequently the distribution of X is known either through its distribution function F or its density function f , and we would similarly like to find the distribution function or density function of Y . This is a difficult problem in general, because as we will see, even simple transformations of variables with simple distributions can lead to variables with complex distributions. We will solve the problem in various special cases. Transformed Variables with Discrete Distributions 2. Suppose that X has a discrete distribution with probability density function f (and hence S is countable). Show that Y has a discrete distribution with probability density function g given by g(y)=∑ f(x), y∈T x∈r −1( y) 3. Suppose that X has a continuous distribution on a subset S⊆ℝ n with probability density function f , and that T is countable. -
The Original Euler's Calculus-Of-Variations Method: Key
Submitted to EJP 1 Jozef Hanc, [email protected] The original Euler’s calculus-of-variations method: Key to Lagrangian mechanics for beginners Jozef Hanca) Technical University, Vysokoskolska 4, 042 00 Kosice, Slovakia Leonhard Euler's original version of the calculus of variations (1744) used elementary mathematics and was intuitive, geometric, and easily visualized. In 1755 Euler (1707-1783) abandoned his version and adopted instead the more rigorous and formal algebraic method of Lagrange. Lagrange’s elegant technique of variations not only bypassed the need for Euler’s intuitive use of a limit-taking process leading to the Euler-Lagrange equation but also eliminated Euler’s geometrical insight. More recently Euler's method has been resurrected, shown to be rigorous, and applied as one of the direct variational methods important in analysis and in computer solutions of physical processes. In our classrooms, however, the study of advanced mechanics is still dominated by Lagrange's analytic method, which students often apply uncritically using "variational recipes" because they have difficulty understanding it intuitively. The present paper describes an adaptation of Euler's method that restores intuition and geometric visualization. This adaptation can be used as an introductory variational treatment in almost all of undergraduate physics and is especially powerful in modern physics. Finally, we present Euler's method as a natural introduction to computer-executed numerical analysis of boundary value problems and the finite element method. I. INTRODUCTION In his pioneering 1744 work The method of finding plane curves that show some property of maximum and minimum,1 Leonhard Euler introduced a general mathematical procedure or method for the systematic investigation of variational problems. -
A Variational Approach to Lagrange Multipliers
JOTA manuscript No. (will be inserted by the editor) A Variational Approach to Lagrange Multipliers Jonathan M. Borwein · Qiji J. Zhu Received: date / Accepted: date Abstract We discuss Lagrange multiplier rules from a variational perspective. This allows us to highlight many of the issues involved and also to illustrate how broadly an abstract version can be applied. Keywords Lagrange multiplier Variational method Convex duality Constrained optimiza- tion Nonsmooth analysis · · · · Mathematics Subject Classification (2000) 90C25 90C46 49N15 · · 1 Introduction The Lagrange multiplier method is fundamental in dealing with constrained optimization prob- lems and is also related to many other important results. There are many different routes to reaching the fundamental result. The variational approach used in [1] provides a deep under- standing of the nature of the Lagrange multiplier rule and is the focus of this survey. David Gale's seminal paper [2] provides a penetrating explanation of the economic meaning of the Lagrange multiplier in the convex case. Consider maximizing the output of an economy with resource constraints. Then the optimal output is a function of the level of resources. It turns out the derivative of this function, if exists, is exactly the Lagrange multiplier for the constrained optimization problem. A Lagrange multiplier, then, reflects the marginal gain of the output function with respect to the vector of resource constraints. Following this observation, if we penalize the resource utilization with a (vector) Lagrange multiplier then the constrained optimization problem can be converted to an unconstrained one. One cannot emphasize enough the importance of this insight. In general, however, an optimal value function for a constrained optimization problem is nei- ther convex nor smooth.