Deriving and Deducing the Equation of the Curve of Quickest Descent
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General Issue
Journal of the British Interplanetary Society VOLUME 72 NO.5 MAY 2019 General issue A REACTION DRIVE Powered by External Dynamic Pressure Jeffrey K. Greason DOCKING WITH ROTATING SPACE SYSTEMS Mark Hempsell MARTIAN DUST: the Formation, Composition, Toxicology, Astronaut Exposure Health Risks and Measures to Mitigate Exposure John Cain THE CONCEPT OF A LOW COST SPACE MISSION to Measure “Big G” Roger Longstaff www.bis-space.com ISSN 0007-084X PUBLICATION DATE: 26 JULY 2019 Submitting papers International Advisory Board to JBIS JBIS welcomes the submission of technical Rachel Armstrong, Newcastle University, UK papers for publication dealing with technical Peter Bainum, Howard University, USA reviews, research, technology and engineering in astronautics and related fields. Stephen Baxter, Science & Science Fiction Writer, UK James Benford, Microwave Sciences, California, USA Text should be: James Biggs, The University of Strathclyde, UK ■ As concise as the content allows – typically 5,000 to 6,000 words. Shorter papers (Technical Notes) Anu Bowman, Foundation for Enterprise Development, California, USA will also be considered; longer papers will only Gerald Cleaver, Baylor University, USA be considered in exceptional circumstances – for Charles Cockell, University of Edinburgh, UK example, in the case of a major subject review. Ian A. Crawford, Birkbeck College London, UK ■ Source references should be inserted in the text in square brackets – [1] – and then listed at the Adam Crowl, Icarus Interstellar, Australia end of the paper. Eric W. Davis, Institute for Advanced Studies at Austin, USA ■ Illustration references should be cited in Kathryn Denning, York University, Toronto, Canada numerical order in the text; those not cited in the Martyn Fogg, Probability Research Group, UK text risk omission. -
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. -
Robert Hooke
? Robert Hooke (1635-1703) Arguably the greatest experimental natural philosopher of the 17th century • His most famous work is Micrographia, in which he coined the term "cell" for a basic biological structure. Biography • Robert Hooke, the son of a clergyman in Freshwater on the Isle of Wight, was born on July 18, 1635. • When he was 13, he left an orphan with a modest inheritance, and entered Westminster School. • Later he earned his way • as a chorister at Christ Church, Oxford, and attended Westminster College, graduating with his master's degree in 1663. Biography • Hooke remained at Oxford, where he became assistant to Robert Boyle. • Together they conducted many experiments on the effects of reduced air pressure, using an air pump that had been designed and constructed by Hooke. Biography… Founded in 1660, the Royal Society is the independent • In 1662 Hooke became scientific academy of the curator of the newly UK, dedicated to promoting founded Royal Society, excellence in science. his duties being to produce three or four significant experimental demonstrations for each weekly meeting of the society. • He was ideally suited for such work, and his career thereafter was immensely active and fertile. Biography… • At Westminster Hooke was said to have acquired mastery of ancient languages, learned to play the organ, 'contrived severall ways of flying', and mastered the first six books of Euclid's Elements. Biography… • 1665 he was appointed Professor of Geometry and carried out astronomical observations, and was also elected FRS. In 1677 he became the Secretary to the Royal Society. Hooke versus Newton • In his Attempt to Prove the Motion of the Earth (1674), he offered a theory of planetary motion based on the correct principle of inertia and a balance between an outward centrifugal force and an inward gravitational attraction to the Sun. -
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. -
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. -
The Evolution of Tower Clock Movements and Their Design Over the Past 1000 Years
The Evolution Of Tower Clock Movements And Their Design Over The Past 1000 Years Mark Frank Copyright 2013 The Evolution Of Tower Clock Movements And Their Design Over The Past 1000 Years TABLE OF CONTENTS Introduction and General Overview Pre-History ............................................................................................... 1. 10th through 11th Centuries ........................................................................ 2. 12th through 15th Centuries ........................................................................ 4. 16th through 17th Centuries ........................................................................ 5. The catastrophic accident of Big Ben ........................................................ 6. 18th through 19th Centuries ........................................................................ 7. 20th Century .............................................................................................. 9. Tower Clock Frame Styles ................................................................................... 11. Doorframe and Field Gate ......................................................................... 11. Birdcage, End-To-End .............................................................................. 12. Birdcage, Side-By-Side ............................................................................. 12. Strap, Posted ............................................................................................ 13. Chair Frame ............................................................................................. -
UTOPIAE Optimisation and Uncertainty Quantification CPD for Teachers of Advanced Higher Physics and Mathematics
UTOPIAE Optimisation and Uncertainty Quantification CPD for Teachers of Advanced Higher Physics and Mathematics Outreach Material Peter McGinty Annalisa Riccardi UTOPIAE, UNIVERSITY OF STRATHCLYDE GLASGOW - UK This research was funded by the European Commission’s Horizon 2020 programme under grant number 722734 First release, October 2018 Contents 1 Introduction ....................................................5 1.1 Who is this for?5 1.2 What are Optimisation and Uncertainty Quantification and why are they im- portant?5 2 Optimisation ...................................................7 2.1 Brachistochrone problem definition7 2.2 How to solve the problem mathematically8 2.3 How to solve the problem experimentally 10 2.4 Examples of Brachistochrone in real life 11 3 Uncertainty Quantification ..................................... 13 3.1 Probability and Statistics 13 3.2 Experiment 14 3.3 Introduction of Uncertainty 16 1. Introduction 1.1 Who is this for? The UTOPIAE Network is committed to creating high quality engagement opportunities for the Early Stage Researchers working within the network and also outreach materials for the wider academic community to benefit from. Optimisation and Uncertainty Quantification, whilst representing the future for a large number of research fields, are relatively unknown disciplines and yet the benefit and impact they can provide for researchers is vast. In order to try to raise awareness, UTOPIAE has created a Continuous Professional Development online resource and training sessions aimed at teachers of Advanced Higher Physics and Mathematics, which is in line with Scotland’s Curriculum for Excellence. Sup- ported by the Glasgow City Council, the STEM Network, and Scottish Schools Education Resource Centre these materials have been published online for teachers to use within a classroom context. -
A Tale of the Cycloid in Four Acts
A Tale of the Cycloid In Four Acts Carlo Margio Figure 1: A point on a wheel tracing a cycloid, from a work by Pascal in 16589. Introduction In the words of Mersenne, a cycloid is “the curve traced in space by a point on a carriage wheel as it revolves, moving forward on the street surface.” 1 This deceptively simple curve has a large number of remarkable and unique properties from an integral ratio of its length to the radius of the generating circle, and an integral ratio of its enclosed area to the area of the generating circle, as can be proven using geometry or basic calculus, to the advanced and unique tautochrone and brachistochrone properties, that are best shown using the calculus of variations. Thrown in to this assortment, a cycloid is the only curve that is its own involute. Study of the cycloid can reinforce the curriculum concepts of curve parameterisation, length of a curve, and the area under a parametric curve. Being mechanically generated, the cycloid also lends itself to practical demonstrations that help visualise these abstract concepts. The history of the curve is as enthralling as the mathematics, and involves many of the great European mathematicians of the seventeenth century (See Appendix I “Mathematicians and Timeline”). Introducing the cycloid through the persons involved in its discovery, and the struggles they underwent to get credit for their insights, not only gives sequence and order to the cycloid’s properties and shows which properties required advances in mathematics, but it also gives a human face to the mathematicians involved and makes them seem less remote, despite their, at times, seemingly superhuman discoveries. -
On a Tautochrone-Related Family of Paths
ENSENANZA˜ REVISTA MEXICANA DE FISICA´ E 56 (2) 227–233 DICIEMBRE 2010 On a tautochrone-related family of paths R. Munoz˜ Universidad Autonoma´ de la Ciudad de Mexico,´ Centro Historico,´ Fray Servando Teresa de Mier 92 y 99, Col. Obrera, Del. Cuauhtemoc,´ Mexico´ D.F., 06080, Mexico,´ e-mail: [email protected] G. Fernandez-Anaya´ Universidad Iberoamericana, Departamento de F´ısica y Matematicas,´ Av. Prolongacion´ Paseo de la Reforma 880, Col. Lomas de Santa Fe, Del. Alvaro Obregon, Mexico´ D.F., 01219, Mexico,´ e-mail: [email protected] Recibido el 26 de julio de 2010; aceptado el 30 de agosto de 2010 An alternative approach to the properties of the tautochrone and brachistochrone curves is used to introduce a family of curves complying with relations where the time of descent is proportional to a fractional power of the height difference. These curves are classified acording with their symmetries. Further properties of these curves are studied. Keywords: Analytical mechanics; Huygens’s isochrone curve; Abel’s mechanical problem. Utilizamos un tratamiento alternativo de las curvas tautocrona y braquistocrona para introducir una familia de curvas que cumplen con relaciones en las que el tiempo de descenso es directamente proporcional a la altura descendida, elevada a un valor fraccionario. Las mencionadas curvas son clasificadas de acuerdo con sus simetr´ıas. Se estudian otras propiedades de dichas curvas. Descriptores: Mecanica´ anal´ıtica; curva isocrona de Huygens; problema mecanico´ de Abel. PACS: 01.55.+b; 02.30Xx; 02.30.Hq; 02.30.Em 1. Introduction cloid. This is in stark contrast with the inclined plane, where the time of descent is proportional to the square root of the In 1658 Christiaan Huygens made public his discovery of height difference: the tautochrone, that is: the path which a point-like parti- T / (¢y)1=2 (2) cle must follow so that the period of its motion comes out to be exactly independent of its amplitude. -
Fun with the Brachistochrone!
Fun With The Brachistochrone! Rhyan Pink Background Defining the problem May 11, 2002 Establish Some Equations Solving for I Don’t . Final Remarks Abstract Acknowledgements In 1646, Johan Bernoulli proposed the general problem: given a wire bent into arbitrary curve, which curve of the infinitely many possibilities yields fastest descent? This path of fastest descent Home Page is called the brachistochrone. The Brachistochrone problem is to find the curve joining two points along which a frictionless bead will descend in minimal time. Earthquake energy propagates through the earth according to Fermat’s Principle of Least Time, and thus earthquake energy will follow the Title Page general shape of this Brachistochrone curve. JJ II 1. Background Earthquakes are a common occurrence around the world, which provides an excellent opportunity for J I geologists to study them. It is well known that there is a higher frequency of earthquakes in some regions versus others, but the precise reasons for why this discrepancy exists is still being studied. Earthquakes Page 1 of 18 are a result of a pressure release as the earth shifts its crust, according to plate tectonics. When two plates interact with one another, a certain amount of shear stress can be tolerated by the geologic Go Back material that composes the subsurface. Once this range of tolerance is surpassed, an earthquake occurs, releasing a tremendous amount of energy that propagates (travels) in the form of elastic waves: body waves and surface waves. See Figure 1. Full Screen Due to the nature of how energy is released in this setting, some distinct aspects of the energy can be observed. -
Johann Bernoulli's Brachistochrone Solution Using Fermat's Principle Of
Eur. J. Phys. 20 (1999) 299–304. Printed in the UK PII: S0143-0807(99)03257-2 Johann Bernoulli’s brachistochrone solution using Fermat’s principle of least time Herman Erlichson Department of Engineering Science and Physics, The College of Staten Island, The City University of New York, Staten Island, NY l0314, USA E-mail: [email protected] Received 8 April 1999 Abstract. Johann Bernoulli’s brachistochrone problem is now three hundred years old. Bernoulli’s solution to the problem he had proposed used the optical analogy of Fermat’s least-time principle. In this analogy a light ray travels between two points in a vertical plane in a medium of continuously varying index of refraction. This solution and connected material are explored in this paper. 1. Introduction It is now three hundred years since Johann Bernoulli challenged the world with his brachistochrone problem. The statement of the problem is deceptively simple. We quote the English translation of Bernoulli’s words from Struik’s excellent book: ‘Let two points A and B be given in a vertical plane. To find the curve that a point M, moving on a path AMB, must follow that, starting from A, it reaches B in the shortest time under its own gravity.’ [1, p 392] This challenge to the world was originally published in the Acta Eruditorum for June 1696. Bernoulli’s own solution to the problem was published in the Acta Eruditorum for May 1697. This solution utilized Fermat’s optical priciple of least time. Fermat’s least-time principle is equivalent to the optical law of refraction. -
Milestones of Direct Variational Calculus and Its Analysis from The
Computer Assisted Methods in Engineering and Science, 25: 141–225, 2018, doi: 10.24423/cames.25.4.2 Copyright © 2018 by Institute of Fundamental Technological Research, Polish Academy of Sciences TWENTY-FIVE YEARS OF THE CAMES Milestones of Direct Variational Calculus and its Analysis from the 17th Century until today and beyond { Mathemat- ics meets Mechanics { with restriction to linear elasticity Milestones of Direct Variational Calculus and its Analysis from the 17th Century until today and beyond – Mathematics meets Mechanics – with restriction to linear elasticity Erwin Stein Leibniz Universit¨atHannover e-mail: [email protected] 142 E. Stein 1 Networked thinking in Computational Sciences 144 2 Eminent scientists in numerical and structural analysis of elasto-mechanics 145 3 The beginning of cybernetic and holistic thinking in philosophy and natural sciences 146 4 Pre-history of the finite element method (FEM) in the 17th to 19th century 147 4.1 Torricelli’s principle of minimum potential energy of a system of rigid bodies under gravity loads in stable static equilibrium . 147 4.2 Galilei’s approximated solution of minimal time of a frictionless down-gliding mass under gravity load . 148 4.3 Snell’s law of light refraction and Fermat’s principle of least time for the optical path length 149 4.4 Johann I Bernoulli’s call for solutions of Galilei’s problem now called brachistochrone problem, in 1696 and the solutions published by Leibniz in 1697 . 150 4.5 Leibniz’s discovery of the kinetic energy of a mass as a conservation quantity . 152 4.6 Leibniz’s draft of a discrete (direct) solution of the brachistochrone problem .