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Leonhard Euler: His Life, the Man, and His Works∗
SIAM REVIEW c 2008 Walter Gautschi Vol. 50, No. 1, pp. 3–33 Leonhard Euler: His Life, the Man, and His Works∗ Walter Gautschi† Abstract. On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, andBerlin—are sketchedandthe principal works identified in more or less chronological order. To convey a flavor of his work andits impact on modernscience, a few of Euler’s memorable contributions are selected anddiscussedinmore detail. Remarks on Euler’s personality, intellect, andcraftsmanship roundout the presentation. Key words. LeonhardEuler, sketch of Euler’s life, works, andpersonality AMS subject classification. 01A50 DOI. 10.1137/070702710 Seh ich die Werke der Meister an, So sehe ich, was sie getan; Betracht ich meine Siebensachen, Seh ich, was ich h¨att sollen machen. –Goethe, Weimar 1814/1815 1. Introduction. It is a virtually impossible task to do justice, in a short span of time and space, to the great genius of Leonhard Euler. All we can do, in this lecture, is to bring across some glimpses of Euler’s incredibly voluminous and diverse work, which today fills 74 massive volumes of the Opera omnia (with two more to come). Nine additional volumes of correspondence are planned and have already appeared in part, and about seven volumes of notebooks and diaries still await editing! We begin in section 2 with a brief outline of Euler’s life, going through the three stations of his life: Basel, St. -
Hamilton's Principle in Continuum Mechanics
Hamilton’s Principle in Continuum Mechanics A. Bedford University of Texas at Austin This document contains the complete text of the monograph published in 1985 by Pitman Publishing, Ltd. Copyright by A. Bedford. 1 Contents Preface 4 1 Mechanics of Systems of Particles 8 1.1 The First Problem of the Calculus of Variations . 8 1.2 Conservative Systems . 12 1.2.1 Hamilton’s principle . 12 1.2.2 Constraints.......................... 15 1.3 Nonconservative Systems . 17 2 Foundations of Continuum Mechanics 20 2.1 Mathematical Preliminaries . 20 2.1.1 Inner Product Spaces . 20 2.1.2 Linear Transformations . 22 2.1.3 Functions, Continuity, and Differentiability . 24 2.1.4 Fields and the Divergence Theorem . 25 2.2 Motion and Deformation . 27 2.3 The Comparison Motion . 32 2.4 The Fundamental Lemmas . 36 3 Mechanics of Continuous Media 39 3.1 The Classical Theories . 40 3.1.1 IdealFluids.......................... 40 3.1.2 ElasticSolids......................... 46 3.1.3 Inelastic Materials . 50 3.2 Theories with Microstructure . 54 3.2.1 Granular Solids . 54 3.2.2 Elastic Solids with Microstructure . 59 2 4 Mechanics of Mixtures 65 4.1 Motions and Comparison Motions of a Mixture . 66 4.1.1 Motions............................ 66 4.1.2 Comparison Fields . 68 4.2 Mixtures of Ideal Fluids . 71 4.2.1 Compressible Fluids . 71 4.2.2 Incompressible Fluids . 73 4.2.3 Fluids with Microinertia . 75 4.3 Mixture of an Ideal Fluid and an Elastic Solid . 83 4.4 A Theory of Mixtures with Microstructure . 86 5 Discontinuous Fields 91 5.1 Singular Surfaces . -
Euler and Chebyshev: from the Sphere to the Plane and Backwards Athanase Papadopoulos
Euler and Chebyshev: From the sphere to the plane and backwards Athanase Papadopoulos To cite this version: Athanase Papadopoulos. Euler and Chebyshev: From the sphere to the plane and backwards. 2016. hal-01352229 HAL Id: hal-01352229 https://hal.archives-ouvertes.fr/hal-01352229 Preprint submitted on 6 Aug 2016 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. EULER AND CHEBYSHEV: FROM THE SPHERE TO THE PLANE AND BACKWARDS ATHANASE PAPADOPOULOS Abstract. We report on the works of Euler and Chebyshev on the drawing of geographical maps. We point out relations with questions about the fitting of garments that were studied by Chebyshev. This paper will appear in the Proceedings in Cybernetics, a volume dedicated to the 70th anniversary of Academician Vladimir Betelin. Keywords: Chebyshev, Euler, surfaces, conformal mappings, cartography, fitting of garments, linkages. AMS classification: 30C20, 91D20, 01A55, 01A50, 53-03, 53-02, 53A05, 53C42, 53A25. 1. Introduction Euler and Chebyshev were both interested in almost all problems in pure and applied mathematics and in engineering, including the conception of industrial ma- chines and technological devices. In this paper, we report on the problem of drawing geographical maps on which they both worked. -
RERL Fact Sheet 1, Community Wind Technology
Renewable Energy Research Laboratory, University of Massachusetts at Amherst Wind Power: Wind Technology Today Community Wind Power on the Community Scale Wind Power Fact Sheet # 1 RERL—MTC Community Wind Wind Power Technology for Fact Sheet Series Communities In collaboration with the Mas- sachusetts Technology Col- This introduction to wind power technology is We also recommend a visit to a modern wind laborative’s Renewable Energy meant to help communities begin considering or power installation – it will answer many of your Trust Fund, the Renewable planning wind power. It focuses on commercial initial questions, including size, noise levels, foot- Energy Research Lab (RERL) and medium-scale wind turbine technology avail- print, and local impact. Some possible field trips able in the United States. are listed on the back page. brings you this series of fact sheets about Wind Power on the community scale: Wind Power Today 1. Technology 2. Performance Wind power is a growing industry, and the technol- 3. Impacts & Issues ogy has changed considerably in recent decades. 4. Siting What would a typical commercial-scale* turbine 5. Resource Assessment installed today look like? 6. Wind Data • Design - 3 blades 7. Permitting - Tubular tower The focus of this series of fact • Hub-height - 164 - 262 ft (50 - 80 m) sheets is medium- • Diameter: - 154 - 262 ft (47 - 80 m) and commercial- • Power ratings available in the US: scale wind. - 660 kW - 1.8 MW * Other scales are discussed below. A wind turbine’s height is usually described as the height of the center of the rotor, or hub. Inside this Edition: Technology Size Ranges What do we mean here when we say “community-scale wind power”? Introduction p. -
Leonhard Euler - Wikipedia, the Free Encyclopedia Page 1 of 14
Leonhard Euler - Wikipedia, the free encyclopedia Page 1 of 14 Leonhard Euler From Wikipedia, the free encyclopedia Leonhard Euler ( German pronunciation: [l]; English Leonhard Euler approximation, "Oiler" [1] 15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[2] He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy. Euler spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes. [3] A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is our teacher in all things," which has also been translated as "Read Portrait by Emanuel Handmann 1756(?) Euler, read Euler, he is the master of us all." [4] Born 15 April 1707 Euler was featured on the sixth series of the Swiss 10- Basel, Switzerland franc banknote and on numerous Swiss, German, and Died Russian postage stamps. The asteroid 2002 Euler was 18 September 1783 (aged 76) named in his honor. He is also commemorated by the [OS: 7 September 1783] Lutheran Church on their Calendar of Saints on 24 St. Petersburg, Russia May – he was a devout Christian (and believer in Residence Prussia, Russia biblical inerrancy) who wrote apologetics and argued Switzerland [5] forcefully against the prominent atheists of his time. -
Coefficient of Restitution Interpreted As Damping in Vibroimpact K
Coefficient of restitution interpreted as damping in vibroimpact Kenneth Hunt, Erskine Crossley To cite this version: Kenneth Hunt, Erskine Crossley. Coefficient of restitution interpreted as damping in vibroimpact. Journal of Applied Mechanics, American Society of Mechanical Engineers, 1975, 10.1115/1.3423596. hal-01333795 HAL Id: hal-01333795 https://hal.archives-ouvertes.fr/hal-01333795 Submitted on 19 Jun 2016 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. Coefficient of Restitution Interpreted as Damping in Vibroimpact K. H. Hunt During impact the relative motion of two bodies is often taken to be simply represented Dean and Professor of Engineering, as half of a damped sine wave, according to the Kelvin-Voigt model. This is shown to be Monash University, logically untenable, for it indicates that the bodies must exert tension on one another Clayton, VIctoria, Australia just before separating. Furthermore, it denotes that the damping energy loss is propor tional to the square of the impactin{f velocity, instead of to its cube, as can be deduced F. R. E. Crossley from Goldsmith's work. A damping term h"i is here introduced; for a sphere impacting Professor, a plate Hertz gives n = 3/2. -
Siméon-Denis Poisson Mathematics in the Service of Science
S IMÉ ON-D E N I S P OISSON M ATHEMATICS I N T H E S ERVICE O F S CIENCE E XHIBITION AT THE MATHEMATICS LIBRARY U NIVE RSIT Y O F I L L I N O I S A T U RBANA - C HAMPAIGN A U G U S T 2014 Exhibition on display in the Mathematics Library of the University of Illinois at Urbana-Champaign 4 August to 14 August 2014 in association with the Poisson 2014 Conference and based on SIMEON-DENIS POISSON, LES MATHEMATIQUES AU SERVICE DE LA SCIENCE an exhibition at the Mathematics and Computer Science Research Library at the Université Pierre et Marie Curie in Paris (MIR at UPMC) 19 March to 19 June 2014 Cover Illustration: Portrait of Siméon-Denis Poisson by E. Marcellot, 1804 © Collections École Polytechnique Revised edition, February 2015 Siméon-Denis Poisson. Mathematics in the Service of Science—Exhibition at the Mathematics Library UIUC (2014) SIMÉON-DENIS POISSON (1781-1840) It is not too difficult to remember the important dates in Siméon-Denis Poisson’s life. He was seventeen in 1798 when he placed first on the entrance examination for the École Polytechnique, which the Revolution had created four years earlier. His subsequent career as a “teacher-scholar” spanned the years 1800-1840. His first publications appeared in the Journal de l’École Polytechnique in 1801, and he died in 1840. Assistant Professor at the École Polytechnique in 1802, he was named Professor in 1806, and then, in 1809, became a professor at the newly created Faculty of Sciences of the Université de Paris. -
Applied Mechanics - 3300008 Unit 1: Introduction 1
MECHANICAL ENGG. DEPT SEMESTER #2 APPLIED MECHANICS - 3300008 UNIT 1: INTRODUCTION 1. Differentiate: 1) Vector quantity and Scalar quantity 2) Kinetics and Kinematics 2. State SI system unit of following quantities: 1) Density 2) Angle 3) Work 4) Power 5) Force 6) Pressure 7) Velocity 8) Torque 3. Name the type of quantities: 1) Density 2) Time 3) Work 4) Energy 5) Force 6) Mass 7) Velocity 8) Speed UNIT 2: COPLANAR CONCURRENT FORCES 1. Explain principle of super position of forces. 2. Explain principle of transmissibility of forces. 3. State and prove lami’s theorem. State its limitation. 4. Explain polygon law of forces. 5. Classify forces. 6. State and Explain law of parallelogram of forces. 7. State and Explain law of triangle of forces. 8. The system shown in figure-1 is in equilibrium. Find unknown forces P and Q. 9. Find magnitude and direction of resultant force for fig.2 Fig.1 Fig. 2 Fig.3 Fig.4 10. A load of 75 KN is hung by means of a rope attached to a hook in horizontal ceiling. What horizontal force should be applied so that rope makes 60o with the ceiling? Page 1 of 27 MECHANICAL ENGG. DEPT SEMESTER #2 11. The boy in a garden holding two chains in his hand which are Hooked with Horizontal steel bar making an angel 65° & other chain with an angle of 55° With the steel bar, if the weight of boy is 55 kg, than find tension developed in both chain. 12. A circular sphere weighing 500 N and having a radius of 200mm hangs by a string AC 400 mm long as shown in Figure 3. -
A Method That Lends Wings
FLASHBACK_Aerodynamics A method that lends wings Mathematician Irmgard Flügge-Lotz was one of the first female researchers in the field of aerodynamics and automatic control. While working at the Kaiser Wilhelm Institute for Flow Research, she succeeded in simplifying the calculations required for aircraft construction. Flügge-Lotz was later appointed the first female Professor of Engineering at Stanford University. In the U.S., her work is still held in high esteem. In Germany, however, she has been all but forgotten. TEXT KATJA ENGEL Goettingen, 1931. Leading flow researcher could only be tested by means of complex Ludwig Prandtl was astonished. His colleague, wind tunnel measurements. Ludwig Prandtl, at 28 just half his age, had just handed him who is generally thought to be the founder of the solution to a mathematical puzzle that no- aircraft aerodynamics, and his team in Goet- body had been able to crack for more than ten tingen carried out pioneering work on the years. This conundrum was on his “menu”, as theoretical description of lift. However, math- he called his list of uncompleted research ematical calculations of his wing theory tasks. The result went down in the history of turned out to be difficult. In 1919, Albert Betz, aerodynamic research together with the a doctoral student in Goettingen who was young researcher's name. The "Lotz method" later to become Prandtl’s successor at the In- makes it possible to calculate the lift on an air- stitute, finally succeeded in the describing lift plane wing with comparative ease. by means of differential equations. However, Prandtl soon made the woman who had so his formulae were too complex for practical impressed him an (unofficial) Head of Depart- use when constructing new profiles. -
Analytical Continuum Mechanics À La Hamilton-Piola: Least Action Principle for Second Gradient Continua and Capillary Fluids Nicolas Auffray, F
Analytical continuum mechanics à la Hamilton-Piola: least action principle for second gradient continua and capillary fluids Nicolas Auffray, F. Dell’Isola, V. Eremeyev, A. Madeo, G.Rosi To cite this version: Nicolas Auffray, F. Dell’Isola, V. Eremeyev, A. Madeo, G. Rosi. Analytical continuum mechanics àla Hamilton-Piola: least action principle for second gradient continua and capillary fluids. Mathematics and Mechanics of Solids, SAGE Publications, 2013. hal-00836085 HAL Id: hal-00836085 https://hal.archives-ouvertes.fr/hal-00836085 Submitted on 21 Jun 2013 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. Analytical continuum mechanics à la Hamilton-Piola: least action principle for second gradient continua and capillary fluids By N. Auffraya, F. dell’Isolab, V. Eremeyevc, A. Madeod and G. Rosie a Université Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-la-Vallée, France bDipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma La Sapienza, Via Eudossiana 18, 00184, Roma, -
Lagrangian Mechanics - Wikipedia, the Free Encyclopedia Page 1 of 11
Lagrangian mechanics - Wikipedia, the free encyclopedia Page 1 of 11 Lagrangian mechanics From Wikipedia, the free encyclopedia Lagrangian mechanics is a re-formulation of classical mechanics that combines Classical mechanics conservation of momentum with conservation of energy. It was introduced by the French mathematician Joseph-Louis Lagrange in 1788. Newton's Second Law In Lagrangian mechanics, the trajectory of a system of particles is derived by solving History of classical mechanics · the Lagrange equations in one of two forms, either the Lagrange equations of the Timeline of classical mechanics [1] first kind , which treat constraints explicitly as extra equations, often using Branches [2][3] Lagrange multipliers; or the Lagrange equations of the second kind , which Statics · Dynamics / Kinetics · Kinematics · [1] incorporate the constraints directly by judicious choice of generalized coordinates. Applied mechanics · Celestial mechanics · [4] The fundamental lemma of the calculus of variations shows that solving the Continuum mechanics · Lagrange equations is equivalent to finding the path for which the action functional is Statistical mechanics stationary, a quantity that is the integral of the Lagrangian over time. Formulations The use of generalized coordinates may considerably simplify a system's analysis. Newtonian mechanics (Vectorial For example, consider a small frictionless bead traveling in a groove. If one is tracking the bead as a particle, calculation of the motion of the bead using Newtonian mechanics) mechanics would require solving for the time-varying constraint force required to Analytical mechanics: keep the bead in the groove. For the same problem using Lagrangian mechanics, one Lagrangian mechanics looks at the path of the groove and chooses a set of independent generalized Hamiltonian mechanics coordinates that completely characterize the possible motion of the bead. -
Active Management of Flap-Edge Trailing Vortices*
4th Flow Control Conference, Seattle, Washington, June 23-26, 2008 Active Management of Flap-Edge Trailing Vortices* David Greenblatt Technion – Israel Institute of Technology, Haifa, Israel Chung-Sheng Yao† NASA Langley Research Center, Hampton VA, USA Stefan Vey,‡ Oliver C. Paschereit¶ Berlin University of Technology, Berlin, Germany Robert Meyer§ German Aerospace Center (DLR), Berlin, Germany The vortex hazard produced by large airliners and increasingly larger airliners entering service, combined with projected rapid increases in the demand for air transportation, is expected to act as a major impediment to increased air traffic capacity. Significant reduction in the vortex hazard is possible, however, by employing active vortex alleviation techniques that reduce the wake severity by dynamically modifying its vortex characteristics, providing that the techniques do not degrade performance or compromise safety and ride quality. With this as background, a series of experiments were performed, initially at NASA Langley Research Center and subsequently at the Berlin University of Technology in collaboration with the German Aerospace Center. The investigations demonstrated the basic mechanism for managing trailing vortices using retrofitted devices that are decoupled from conventional control surfaces. The basic premise for managing vortices advanced here is rooted in the erstwhile forgotten hypothesis of Albert Betz, as extended and verified ingeniously by Coleman duPont Donaldson and his collaborators. Using these devices, vortices may be perturbed at arbitrarily long wavelengths down to wavelengths less than a typical airliner wingspan and the oscillatory loads on the wings, and hence the vehicle, are small. Significant flexibility in the specific device has been demonstrated using local passive and active separation control as well as local circulation control via Gurney flaps.