Optimization in the History of Science
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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 ........................ -
Network Map of Knowledge And
Humphry Davy George Grosz Patrick Galvin August Wilhelm von Hofmann Mervyn Gotsman Peter Blake Willa Cather Norman Vincent Peale Hans Holbein the Elder David Bomberg Hans Lewy Mark Ryden Juan Gris Ian Stevenson Charles Coleman (English painter) Mauritz de Haas David Drake Donald E. Westlake John Morton Blum Yehuda Amichai Stephen Smale Bernd and Hilla Becher Vitsentzos Kornaros Maxfield Parrish L. Sprague de Camp Derek Jarman Baron Carl von Rokitansky John LaFarge Richard Francis Burton Jamie Hewlett George Sterling Sergei Winogradsky Federico Halbherr Jean-Léon Gérôme William M. Bass Roy Lichtenstein Jacob Isaakszoon van Ruisdael Tony Cliff Julia Margaret Cameron Arnold Sommerfeld Adrian Willaert Olga Arsenievna Oleinik LeMoine Fitzgerald Christian Krohg Wilfred Thesiger Jean-Joseph Benjamin-Constant Eva Hesse `Abd Allah ibn `Abbas Him Mark Lai Clark Ashton Smith Clint Eastwood Therkel Mathiassen Bettie Page Frank DuMond Peter Whittle Salvador Espriu Gaetano Fichera William Cubley Jean Tinguely Amado Nervo Sarat Chandra Chattopadhyay Ferdinand Hodler Françoise Sagan Dave Meltzer Anton Julius Carlson Bela Cikoš Sesija John Cleese Kan Nyunt Charlotte Lamb Benjamin Silliman Howard Hendricks Jim Russell (cartoonist) Kate Chopin Gary Becker Harvey Kurtzman Michel Tapié John C. Maxwell Stan Pitt Henry Lawson Gustave Boulanger Wayne Shorter Irshad Kamil Joseph Greenberg Dungeons & Dragons Serbian epic poetry Adrian Ludwig Richter Eliseu Visconti Albert Maignan Syed Nazeer Husain Hakushu Kitahara Lim Cheng Hoe David Brin Bernard Ogilvie Dodge Star Wars Karel Capek Hudson River School Alfred Hitchcock Vladimir Colin Robert Kroetsch Shah Abdul Latif Bhittai Stephen Sondheim Robert Ludlum Frank Frazetta Walter Tevis Sax Rohmer Rafael Sabatini Ralph Nader Manon Gropius Aristide Maillol Ed Roth Jonathan Dordick Abdur Razzaq (Professor) John W. -
Appendix a Short Course in Taylor Series
Appendix A Short Course in Taylor Series The Taylor series is mainly used for approximating functions when one can identify a small parameter. Expansion techniques are useful for many applications in physics, sometimes in unexpected ways. A.1 Taylor Series Expansions and Approximations In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It is named after the English mathematician Brook Taylor. If the series is centered at zero, the series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin. It is common practice to use a finite number of terms of the series to approximate a function. The Taylor series may be regarded as the limit of the Taylor polynomials. A.2 Definition A Taylor series is a series expansion of a function about a point. A one-dimensional Taylor series is an expansion of a real function f(x) about a point x ¼ a is given by; f 00ðÞa f 3ðÞa fxðÞ¼faðÞþf 0ðÞa ðÞþx À a ðÞx À a 2 þ ðÞx À a 3 þÁÁÁ 2! 3! f ðÞn ðÞa þ ðÞx À a n þÁÁÁ ðA:1Þ n! © Springer International Publishing Switzerland 2016 415 B. Zohuri, Directed Energy Weapons, DOI 10.1007/978-3-319-31289-7 416 Appendix A: Short Course in Taylor Series If a ¼ 0, the expansion is known as a Maclaurin Series. Equation A.1 can be written in the more compact sigma notation as follows: X1 f ðÞn ðÞa ðÞx À a n ðA:2Þ n! n¼0 where n ! is mathematical notation for factorial n and f(n)(a) denotes the n th derivation of function f evaluated at the point a. -
Introduction to the Modern Calculus of Variations
MA4G6 Lecture Notes Introduction to the Modern Calculus of Variations Filip Rindler Spring Term 2015 Filip Rindler Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom [email protected] http://www.warwick.ac.uk/filiprindler Copyright ©2015 Filip Rindler. Version 1.1. Preface These lecture notes, written for the MA4G6 Calculus of Variations course at the University of Warwick, intend to give a modern introduction to the Calculus of Variations. I have tried to cover different aspects of the field and to explain how they fit into the “big picture”. This is not an encyclopedic work; many important results are omitted and sometimes I only present a special case of a more general theorem. I have, however, tried to strike a balance between a pure introduction and a text that can be used for later revision of forgotten material. The presentation is based around a few principles: • The presentation is quite “modern” in that I use several techniques which are perhaps not usually found in an introductory text or that have only recently been developed. • For most results, I try to use “reasonable” assumptions, not necessarily minimal ones. • When presented with a choice of how to prove a result, I have usually preferred the (in my opinion) most conceptually clear approach over more “elementary” ones. For example, I use Young measures in many instances, even though this comes at the expense of a higher initial burden of abstract theory. • Wherever possible, I first present an abstract result for general functionals defined on Banach spaces to illustrate the general structure of a certain result. -
Staying Optimistic: the Trials and Tribulations of Leibnizian Optimism
Strickland, Lloyd 2019 Staying Optimistic: The Trials and Tribulations of Leibnizian Optimism. Journal of Modern Philosophy, 1(1): 3, pp. 1–21. DOI: https://doi.org/10.32881/jomp.3 RESEARCH Staying Optimistic: The Trials and Tribulations of Leibnizian Optimism Lloyd Strickland Manchester Metropolitan University, GB [email protected] The oft-told story of Leibniz’s doctrine of the best world, or optimism, is that it enjoyed a great deal of popularity in the eighteenth century until the massive earthquake that struck Lisbon on 1 November 1755 destroyed its support. Despite its long history, this story is nothing more than a commentators’ fiction that has become accepted wisdom not through sheer weight of evidence but through sheer frequency of repetition. In this paper we shall examine the reception of Leibniz’s doctrine of the best world in the eighteenth century in order to get a clearer understanding of what its fate really was. As we shall see, while Leibniz’s doctrine did win a good number of adherents in the 1720s and 1730s, especially in Germany, support for it had largely dried up by the mid-1740s; moreover, while opponents of Leibniz’s doctrine were few and far between in the 1710s and 1720s, they became increasing vocal in the 1730s and afterwards, between them producing an array of objections that served to make Leibnizian optimism both philosophically and theologically toxic years before the Lisbon earthquake struck. Keywords: Leibniz; Optimism; Best world; Lisbon earthquake; Evil; Wolff The oft-told story of Leibniz’s doctrine of the best world, or optimism, is that it enjoyed a great deal of popularity in the eighteenth century until the massive earthquake that struck Lisbon on 1 November 1755 destroyed its support. -
Pre-Darwinian Thinking, the Voyage of the Beagle, and the Origin of Species
Pre-Darwinian thinking, the voyage of the Beagle, and the Origin of Species How did life originate? What is responsible for the spectacular diversity that we see now? These are questions that have occupied numerous people for all of recorded history. Indeed, many independent mythological traditions have stories that discuss origins, and some of those are rather creative. For example, in Norse mythology the world was created out of the body of a frost giant! Some early attempts at scientific or philosophical discussion of life were apparent in Greece. For example, Thales (the earliest of the identified Greek philosophers) argued that everything stemmed ultimately from water. He therefore had a somewhat vague idea that descent with modification was possible, since things had to diversify from a common origin. Aristotle suggested that in every thing is a desire to move from lower to higher forms, and ultimately to the divine. Anaximander might have come closest to our modern conception: he proposed that humans originated from other animals, based on the observation that babies need care for such a long time that if the first humans had started like that, they would not have survived. Against this, however, is the observation that in nature, over human-scale observation times, very little seems to change about life as we can see it with our unaided eyes. Sure, the offspring of an individual animal aren’t identical to it, but puppies grow up to be dogs, not cats. Even animals that have very short generational times appear not to change sub- stantially: one fruit fly is as good as another. -
Alexis Claude Clairaut
Alexis Claude Clairaut Born: 7 May 1713 in Paris, France Died: 17 May 1765 in Paris, France Alexis Clairaut's father, Jean-Baptiste Clairaut, taught mathematics in Paris and showed his quality by being elected to the Berlin Academy. Alexis's mother, Catherine Petit, had twenty children although only Alexis survived to adulthood. Jean-Baptiste Clairaut educated his son at home and set unbelievably high standards. Alexis used Euclid's Elements while learning to read and by the age of nine he had mastered the excellent mathematics textbook of Guisnée Application de l'algèbre à la géométrie which provided a good introduction to the differential and integral calculus as well as analytical geometry. In the following year, Clairaut went on to study de L'Hôpital's books, in particular his famous text Analyse des infiniment petits pour l'intelligence des lignes courbes. Few people have read their first paper to an academy at the age of 13, but this was the incredible achievement of Clairaut's in 1726 when he read his paper Quatre problèmes sur de nouvelles courbes to the Paris Academy. Although we have already noted that Clairaut was the only one of twenty children of his parents to reach adulthood, he did have a younger brother who, at the age of 14, read a mathematics paper to the Academy in 1730. This younger brother died in 1732 at the age of 16. Clairaut began to undertake research on double curvature curves which he completed in 1729. As a result of this work he was proposed for membership of the Paris Academy on 4 September 1729 but the king did not confirm his election until 1731. -
Calculus of Variations
MIT OpenCourseWare http://ocw.mit.edu 16.323 Principles of Optimal Control Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 16.323 Lecture 5 Calculus of Variations • Calculus of Variations • Most books cover this material well, but Kirk Chapter 4 does a particularly nice job. • See here for online reference. x(t) x*+ αδx(1) x*- αδx(1) x* αδx(1) −αδx(1) t t0 tf Figure by MIT OpenCourseWare. Spr 2008 16.323 5–1 Calculus of Variations • Goal: Develop alternative approach to solve general optimization problems for continuous systems – variational calculus – Formal approach will provide new insights for constrained solutions, and a more direct path to the solution for other problems. • Main issue – General control problem, the cost is a function of functions x(t) and u(t). � tf min J = h(x(tf )) + g(x(t), u(t), t)) dt t0 subject to x˙ = f(x, u, t) x(t0), t0 given m(x(tf ), tf ) = 0 – Call J(x(t), u(t)) a functional. • Need to investigate how to find the optimal values of a functional. – For a function, we found the gradient, and set it to zero to find the stationary points, and then investigated the higher order derivatives to determine if it is a maximum or minimum. – Will investigate something similar for functionals. June 18, 2008 Spr 2008 16.323 5–2 • Maximum and Minimum of a Function – A function f(x) has a local minimum at x� if f(x) ≥ f(x �) for all admissible x in �x − x�� ≤ � – Minimum can occur at (i) stationary point, (ii) at a boundary, or (iii) a point of discontinuous derivative. -
Calculus of Variations Raju K George, IIST
Calculus of Variations Raju K George, IIST Lecture-1 In Calculus of Variations, we will study maximum and minimum of a certain class of functions. We first recall some maxima/minima results from the classical calculus. Maxima and Minima Let X and Y be two arbitrary sets and f : X Y be a well-defined function having domain → X and range Y . The function values f(x) become comparable if Y is IR or a subset of IR. Thus, optimization problem is valid for real valued functions. Let f : X IR be a real valued → function having X as its domain. Now x X is said to be maximum point for the function f if 0 ∈ f(x ) f(x) x X. The value f(x ) is called the maximum value of f. Similarly, x X is 0 ≥ ∀ ∈ 0 0 ∈ said to be a minimum point for the function f if f(x ) f(x) x X and in this case f(x ) is 0 ≤ ∀ ∈ 0 the minimum value of f. Sufficient condition for having maximum and minimum: Theorem (Weierstrass Theorem) Let S IR and f : S IR be a well defined function. Then f will have a maximum/minimum ⊆ → under the following sufficient conditions. 1. f : S IR is a continuous function. → 2. S IR is a bound and closed (compact) subset of IR. ⊂ Note that the above conditions are just sufficient conditions but not necessary. Example 1: Let f : [ 1, 1] IR defined by − → 1 x = 0 f(x)= − x x = 0 | | 6 1 −1 +1 −1 Obviously f(x) is not continuous at x = 0. -
Calculus of Variations
Calculus of Variations The biggest step from derivatives with one variable to derivatives with many variables is from one to two. After that, going from two to three was just more algebra and more complicated pictures. Now the step will be from a finite number of variables to an infinite number. That will require a new set of tools, yet in many ways the techniques are not very different from those you know. If you've never read chapter 19 of volume II of the Feynman Lectures in Physics, now would be a good time. It's a classic introduction to the area. For a deeper look at the subject, pick up MacCluer's book referred to in the Bibliography at the beginning of this book. 16.1 Examples What line provides the shortest distance between two points? A straight line of course, no surprise there. But not so fast, with a few twists on the question the result won't be nearly as obvious. How do I measure the length of a curved (or even straight) line? Typically with a ruler. For the curved line I have to do successive approximations, breaking the curve into small pieces and adding the finite number of lengths, eventually taking a limit to express the answer as an integral. Even with a straight line I will do the same thing if my ruler isn't long enough. Put this in terms of how you do the measurement: Go to a local store and purchase a ruler. It's made out of some real material, say brass. -
MATHEMATICAL INDUCTION SEQUENCES and SERIES
MISS MATHEMATICAL INDUCTION SEQUENCES and SERIES John J O'Connor 2009/10 Contents This booklet contains eleven lectures on the topics: Mathematical Induction 2 Sequences 9 Series 13 Power Series 22 Taylor Series 24 Summary 29 Mathematician's pictures 30 Exercises on these topics are on the following pages: Mathematical Induction 8 Sequences 13 Series 21 Power Series 24 Taylor Series 28 Solutions to the exercises in this booklet are available at the Web-site: www-history.mcs.st-andrews.ac.uk/~john/MISS_solns/ 1 Mathematical induction This is a method of "pulling oneself up by one's bootstraps" and is regarded with suspicion by non-mathematicians. Example Suppose we want to sum an Arithmetic Progression: 1+ 2 + 3 +...+ n = 1 n(n +1). 2 Engineers' induction € Check it for (say) the first few values and then for one larger value — if it works for those it's bound to be OK. Mathematicians are scornful of an argument like this — though notice that if it fails for some value there is no point in going any further. Doing it more carefully: We define a sequence of "propositions" P(1), P(2), ... where P(n) is "1+ 2 + 3 +...+ n = 1 n(n +1)" 2 First we'll prove P(1); this is called "anchoring the induction". Then we will prove that if P(k) is true for some value of k, then so is P(k + 1) ; this is€ c alled "the inductive step". Proof of the method If P(1) is OK, then we can use this to deduce that P(2) is true and then use this to show that P(3) is true and so on. -
The Trisectrix of Maclaurin, That Provides One Way of Geometrically Trisecting an Angle Exactly
The Maclaurin Trisectrix Justin Seago December 8, 2008 Colin Maclaurin (1698-1746) • Scottish mathematician • Entered University of Glasgow at the age of 11 and received his M.A. at 14. [2] • Elected professor of mathematics at the University of Aberdeen at the age of 19, making him the youngest professor in history until March 2008. [4] • Made use of the form of a Taylor Series that bears his name in A Treatise on Fluxions (1742) wherein he defended Newton’s calculus. [2] [3] Geometry: Trisecting an Angle Imagine two lines separated by an arbitrary angle (θ). Now, our objective is to precisely trisect this angle; θ that is, geometrically divide the angle into three equal angles (θ/3). If we made a line spanning the angle and divided it a into three equal lengths, would there be three equal angles? θ/3? θ/3? a θ/3? a Trisection of General Angles Here GeoGebra is employed to model the attempted trisection. If θ is taken to be the angle we wish to trisect, and θ = 96.29°, then θ/3 should be approximately 32.01° As you can see in the screen capture from GeoGebra at right, trisecting a line spanning the arc of θ gives at best a rough approximation of θ/3. Trisection of General Angles Here is a second attempt. Although this greatly improves our approximation from the previous method, drawing this accurately is difficult and the resulting angles are still not exactly θ/3. Maclaurin’s Solution In 1742 Colin Maclaurin discovered a curve, now called the Maclaurin Trisectrix or The Trisectrix of Maclaurin, that provides one way of geometrically trisecting an angle exactly.