Towards Energy Principles in the 18Th and 19Th Century – from D’Alembert to Gauss

Total Page:16

File Type:pdf, Size:1020Kb

Towards Energy Principles in the 18Th and 19Th Century – from D’Alembert to Gauss Towards Energy Principles in the 18th and 19th Century { From D'Alembert to Gauss Ekkehard Ramm, Universit¨at Stuttgart The present contribution describes the evolution of extremum principles in mechanics in the 18th and the first half of the 19th century. First the development of the 'Principle of Least Action' is recapitulated [1]: Maupertuis' (1698-1759) hypothesis that for any change in nature there is a quantity for this change, denoted as 'action', which is a minimum (1744/46); S. Koenig's contribution in 1750 against Maupertuis, president of the Prussian Academy of Science, delivering a counter example that a maximum may occur as well and most importantly presenting a copy of a letter written by Leibniz already in 1707 which describes Maupertuis' general principle but allowing for a minimum or maximum; Euler (1707-1783) heavily defended Maupertuis in this priority rights although he himself had discovered the principle before him. Next we refer to Jean Le Rond d'Alembert (1717-1783), member of the Paris Academy of Science since 1741. He described his principle of mechanics in his 'Trait´ede dynamique' in 1743. It is remarkable that he was considered more a mathematician rather than a physicist; he himself 'believed mechanics to be based on metaphysical principles and not on experimental evidence' [2]. Ne- vertheless D'Alembert's Principle, expressing the dynamic equilibrium as the kinetic extension of the principle of virtual work, became in its Lagrangian ver- sion one of the most powerful contributions in mechanics. Briefly Hamilton's Principle, denoted as 'Law of Varying Action' by Sir William Rowan Hamilton (1805-1865), as the integral counterpart to d'Alembert's differential equation is also mentioned. We then describe in detail the 'Principle of Least Constraint' by Carl Friedrich Gauss (1777-1855), published in the Crelle Journal [3] which still exists today. The paper had only four pages with very little mathematics. Gauss defined the square of the differences between free and constrained accelerations for all masses as measure which should lead to a minimum constraint. Thus he rein- terpreted d'Alembert's principle changing it into a minimum principle [4]. The principle has a direct analogy to his famous least square approximation. The publication was his only work on this subject. The Principles of Virtual Work (Displacements) and Complementary Work (Forces) became major building blocks in mechanics; they are related to the Principles of Minimum of Total Potential Energy (Green-Dirichlet) and Com- plementary Energy (Menabrea-Castilliano) once a potential exists. Furthermore mixed principles have been derived in the 20th century such as the Hellinger- Reissner or the Hu-Washizu Principle which became important starting points in Finite-Element Technology. 1 [1] I Szabo, Geschichte der mechanischen Prinzipien und ihrer wichtigsten Anwendungen, Birkh¨auser, Basel 1979 [2] JJ O'Connor and EF Robertson, Jean Le Rond D'Alembert in MacTutor History of Mathematics, Oct 1998 http://www-history.mcs.st-andrews.ac.uk/Biographies/D'Alembert.html [3] CF Gauss, Uber¨ ein neues allgemeines Grundgesetz der Mechanik, Jour- nal fur¨ die reine und angew. Mathematik, herausg. v. CRELLE, Band IV (1829), 232{235 [4] C Lanczos, The Variational Principles of Mechanics, University of To- ronto Press, 3rd edition 1966, Section 8,106{110 2.
Recommended publications
  • Analogy in William Rowan Hamilton's New Algebra
    Technical Communication Quarterly ISSN: 1057-2252 (Print) 1542-7625 (Online) Journal homepage: https://www.tandfonline.com/loi/htcq20 Analogy in William Rowan Hamilton's New Algebra Joseph Little & Maritza M. Branker To cite this article: Joseph Little & Maritza M. Branker (2012) Analogy in William Rowan Hamilton's New Algebra, Technical Communication Quarterly, 21:4, 277-289, DOI: 10.1080/10572252.2012.673955 To link to this article: https://doi.org/10.1080/10572252.2012.673955 Accepted author version posted online: 16 Mar 2012. Published online: 16 Mar 2012. Submit your article to this journal Article views: 158 Citing articles: 1 View citing articles Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=htcq20 Technical Communication Quarterly, 21: 277–289, 2012 Copyright # Association of Teachers of Technical Writing ISSN: 1057-2252 print/1542-7625 online DOI: 10.1080/10572252.2012.673955 Analogy in William Rowan Hamilton’s New Algebra Joseph Little and Maritza M. Branker Niagara University This essay offers the first analysis of analogy in research-level mathematics, taking as its case the 1837 treatise of William Rowan Hamilton. Analogy spatialized Hamilton’s key concepts—knowl- edge and time—in culturally familiar ways, creating an effective landscape for thinking about the new algebra. It also structurally aligned his theory with the real number system so his objects and operations would behave customarily, thus encompassing the old algebra while systematically bringing into existence the new. Keywords: algebra, analogy, mathematics, William Rowan Hamilton INTRODUCTION Studies of analogy in technical discourse have made important strides in the 30 years since Lakoff and Johnson (1980) ushered in the cognitive linguistic turn.
    [Show full text]
  • 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 .
    [Show full text]
  • Principle of Virtual Work
    Chapter 1 Principle of virtual work 1.1 Constraints and degrees of freedom The number of degrees of freedom of a system is equal to the number of variables required to describe the state of the system. For instance: • A particle constrained to move along the x axis has one degree of freedom, the position x on this axis. • A particle constrained to the surface of the earth has two degrees of freedom, longitude and latitude. • A wheel rotating on a fixed axle has one degree of freedom, the angle of rotation. • A solid body in free space has six degrees of freedom: a particular atom in the body can move in three dimensions, which accounts for three degrees of freedom; another atom can move on a sphere with the first particle at its center for two additional degrees of freedom; any other atom can move in a circle about the line passing through the first two atoms. No other independent motion of the body is possible. • N atoms moving freely in three-dimensional space collectively have 3N degrees of freedom. 1.1.1 Holonomic constraints Suppose a mass is constrained to move in a circle of radius R in the x-y plane. Without this constraint it could move freely over this plane. Such a constraint could be expressed by the equation for a circle, x2 + y2 = R2. A better way to represent this constraint is F (x; y) = x2 + y2 − R2 = 0: (1.1.1) 1 CHAPTER 1. PRINCIPLE OF VIRTUAL WORK 2 As we shall see, this constraint may be useful when expressed in differential form: @F @F dF = dx + dy = 2xdx + 2ydy = 0: (1.1.2) @x @y A constraint that can be represented by setting to zero a function of the variables representing the configuration of a system (e.g., the x and y locations of a mass moving in a plane) is called holonomic.
    [Show full text]
  • Virtual Work
    MEAM 535 Principle of Virtual Work Aristotle Galileo (1594) Bernoulli (1717) Lagrange (1788) 1. Start with static equilibrium of holonomic system of N particles 2. Extend to rigid bodies 3. Incorporate inertial forces for dynamic analysis 4. Apply to nonholonomic systems University of Pennsylvania 1 MEAM 535 Virtual Work Key Ideas (a) Fi Virtual displacement e2 Small Consistent with constraints Occurring without passage of time rPi Applied forces (and moments) Ignore constraint forces Static equilibrium e Zero acceleration, or O 1 Zero mass Every point, Pi, is subject to The virtual work is the work done by a virtual displacement: . e3 the applied forces. N n generalized coordinates, qj (a) Pi δW = ∑[Fi ⋅δr ] i=1 University of Pennsylvania 2 € MEAM 535 Example: Particle in a slot cut into a rotating disk Angular velocity Ω constant Particle P constrained to be in a radial slot on the rotating disk P F r How do describe virtual b2 Ω displacements of the particle P? b1 O No. of degrees of freedom in A? Generalized coordinates? B Velocity of P in A? a2 What is the virtual work done by the force a1 F=F1b1+F2b2 ? University of Pennsylvania 3 MEAM 535 Example l Applied forces G=τ/2r B F acting at P Q r φ θ m F G acting at Q P (assume no gravity) Constraint forces x All joint reaction forces Single degree of freedom Generalized coordinate, θ Motion of particles P and Q can be described by the generalized coordinate θ University of Pennsylvania 4 MEAM 535 Static Equilibrium Implies Zero Virtual Work is Done Forces Forces that do
    [Show full text]
  • 2012 Summer Workshop, College of the Holy Cross Foundational Mathematics Concepts for the High School to College Transition
    2012 Summer Workshop, College of the Holy Cross Foundational Mathematics Concepts for the High School to College Transition Day 9 { July 23, 2012 Summary of Graphs: Along the way, you have used several concepts that arise in the area of mathematics known as discrete mathematics or combinatorics. Here is a list of them: • Going from Amherst to the Mass Maritime academy is the equivalent of moving along a collection of edges in order from one vertex to another so that the vertex at the end of one edge is the vertex of the beginning of the next. This is called a path. • Starting at Worcester and ending at Worcester gives a path that starts at one vertex and ends at the same vertex. This is called a circuit or cycle. • A route that allows you to visit every school is called a Hamiltonian path (if it has a different start and end points) or a Hamiltonian circuit (if it has the same starting and ending point). These are named after the 19th century British physicist, astronomer, and mathematician William Rowan Hamilton. Among other things, he is known for inventing his own system of numbers (can you imagine that!), called quarternions, which are important in mathematics and physics. Figure 1: The bridges of Konigsberg, Prussia. (Wikipedia, entry for Leonhard Euler.) • A route that allows you to drive every road between schools once is called an Eulerian path (if it has a different starting and ending points) or a Eulerian circuit (if it has the same start and end point). These are named after the 18th century German mathematician and physicist Leonhard Euler.
    [Show full text]
  • 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.
    [Show full text]
  • Principle of Virtual Work
    Principle of Virtual Work Degrees of Freedom Associated with the concept of the lumped-mass approximation is the idea of the NUMBER OF DEGREES OF FREEDOM. This can be defined as “the number of independent co-ordinates required to specify the configuration of the system”. The word “independent” here implies that there is no fixed relationship between the co- ordinates, arising from geometric constraints. Modelling of Automotive Systems 1 Degrees of Freedom of Special Systems A particle in free motion in space has 3 degrees of freedom z particle in free motion in space r has 3 degrees of freedom y x 3 If we introduce one constraint – e.g. r is fixed then the number of degrees of freedom reduces to 2. note generally: no. of degrees of freedom = no. of co-ordinates –no. of equations of constraint Modelling of Automotive Systems 2 Rigid Body This has 6 degrees of freedom y 3 translation P2 P1 3 rotation P3 . x 3 e.g. for partials P1, P2 and P3 we have 3 x 3 = 9 co-ordinates but the distances between these particles are fixed – for a rigid body – thus there are 3 equations of constraint. The no. of degrees of freedom = no. of co-ordinates (9) - no. of equations of constraint (3) = 6. Modelling of Automotive Systems 3 Formulation of the Equations of Motion Two basic approaches: 1. application of Newton’s laws of motion to free-body diagrams Disadvantages of Newton’s law approach are that we need to deal with vector quantities – force and displacement. thus we need to resolve in two or three dimensions – choice of method of resolution needs to be made.
    [Show full text]
  • Learning the Virtual Work Method in Statics: What Is a Compatible Virtual Displacement?
    2006-823: LEARNING THE VIRTUAL WORK METHOD IN STATICS: WHAT IS A COMPATIBLE VIRTUAL DISPLACEMENT? Ing-Chang Jong, University of Arkansas Ing-Chang Jong serves as Professor of Mechanical Engineering at the University of Arkansas. He received a BSCE in 1961 from the National Taiwan University, an MSCE in 1963 from South Dakota School of Mines and Technology, and a Ph.D. in Theoretical and Applied Mechanics in 1965 from Northwestern University. He was Chair of the Mechanics Division, ASEE, in 1996-97. His research interests are in mechanics and engineering education. Page 11.878.1 Page © American Society for Engineering Education, 2006 Learning the Virtual Work Method in Statics: What Is a Compatible Virtual Displacement? Abstract Statics is a course aimed at developing in students the concepts and skills related to the analysis and prediction of conditions of bodies under the action of balanced force systems. At a number of institutions, learning the traditional approach using force and moment equilibrium equations is followed by learning the energy approach using the virtual work method to enrich the learning of students. The transition from the traditional approach to the energy approach requires learning several related key concepts and strategy. Among others, compatible virtual displacement is a key concept, which is compatible with what is required in the virtual work method but is not commonly recognized and emphasized. The virtual work method is initially not easy to learn for many people. It is surmountable when one understands the following: (a) the proper steps and strategy in the method, (b) the displacement center, (c) some basic geometry, and (d ) the radian measure formula to compute virtual displacements.
    [Show full text]
  • Leonhard Euler's Elastic Curves Author(S): W
    Leonhard Euler's Elastic Curves Author(s): W. A. Oldfather, C. A. Ellis and Donald M. Brown Source: Isis, Vol. 20, No. 1 (Nov., 1933), pp. 72-160 Published by: The University of Chicago Press on behalf of The History of Science Society Stable URL: http://www.jstor.org/stable/224885 Accessed: 10-07-2015 18:15 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. The University of Chicago Press and The History of Science Society are collaborating with JSTOR to digitize, preserve and extend access to Isis. http://www.jstor.org This content downloaded from 128.138.65.63 on Fri, 10 Jul 2015 18:15:50 UTC All use subject to JSTOR Terms and Conditions LeonhardEuler's ElasticCurves (De Curvis Elasticis, Additamentum I to his Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Lausanne and Geneva, 1744). Translated and Annotated by W. A. OLDFATHER, C. A. ELLIS, and D. M. BROWN PREFACE In the fall of I920 Mr. CHARLES A. ELLIS, at that time Professor of Structural Engineering in the University of Illinois, called my attention to the famous appendix on elastic curves by LEONHARD EULER, which he felt might well be made available in an English translationto those students of structuralengineering who were interested in the classical treatises which constitute landmarks in the history of this ever increasingly important branch of scientific and technical achievement.
    [Show full text]
  • ON THEOREMS of CENTRAL FORCES by William Rowan Hamilton
    ON THEOREMS OF CENTRAL FORCES By William Rowan Hamilton (Proceedings of the Royal Irish Academy, 3 (1847), pp. 308–309.) Edited by David R. Wilkins 2000 On Theorems of Central Forces. By Sir William R. Hamilton. Communicated November 30, 1846. [Proceedings of the Royal Irish Academy, vol. 3 (1847), pp. 308–309.] Sir William R. Hamilton stated the following theorems of central forces, which he had proved by his calculus of quaternions, but which, as he remarked, might be also deduced from principles more elementary. If a body be attracted to a fixed point, with a force which varies directly as the distance from that point, and inversely as the cube of the distance from a fixed plane, the body will describe a conic section, of which the plane intersects the fixed plane in a straight line, which is the polar of the fixed point with respect to the conic section. And in like manner, if a material point be obliged to remain upon the surface of a given sphere, and be acted on by a force, of which the tangential component is constantly directed (along the surface) towards a fixed point or pole upon that surface, and varies directly as the sine of the arcual distance from that pole, and inversely as the cube of the sine of the arcual distance from a fixed great circle; then the material point will describe a spherical conic, with respect to which the fixed great circle will be the polar of the fixed point. Thus, a spherical conic would be described by a heavy point upon a sphere, if the vertical accelerating force were to vary inversely as the cube of the perpendicular and linear distance from a fixed plane passing through the centre.
    [Show full text]
  • Verge 5 Barker.Pdf (420.0Kb)
    Verge 5 Barker 1 by Daniel Barker t2 J = ∫ Ldt t1 ∂L ∂L − d = 0 dt & ∂qi ∂qi Verge 5 Barker 2 Finding the path of motion of a particle is a fundamental physical problem. Inside an inertial frame – a coordinate system that moves with constant velocity – the motion of a system is described by Newton’s Second Law: F = p& , where F is the total force acting on the particle and p& is the time derivative of the particle’s momentum. 1 Provided that the particle’s motion is not complicated and rectangular coordinates are used, then the equations of motion are fairly easy to obtain. In this case, the equations of motion are analytically solvable and the path of the particle can be found using matrix techniques. 2 Unfortunately, situations arise where it is difficult or impossible to obtain an explicit expression for all forces acting on a system. Therefore, a different approach to mechanics is desirable in order to circumvent the difficulties encountered when applying Newton’s laws. 3 The primary obstacle to finding equations of motion using the Newtonian technique is the vector nature of F. We would like to use an alternate formulation of mechanics that uses scalar quantities to derive the equations of motion. Lagrangian mechanics is one such formulation, which is based on Hamilton’s variational principle instead on Newton’s Second Law. 4 Hamilton’s principle – published in 1834 by William Rowan Hamilton – is a mathematical statement of the philosophical belief that the laws of nature obey a principle of economy. Under such a principle, particles follow paths that are extrema for some associated physical quantities.
    [Show full text]
  • Application of Principle of Virtual Work to Find Displacement in Statically Indeterminate Structures
    CE474 – Structural Analysis II Application of Principle of Virtual Work to Find Displacement in Statically Indeterminate Structures Example: Find B , the vertical displacement at B. Consider flexural response only; assume EI is constant. First of all, we need to find the curvature distribution in this statically indeterminate to 1st degree propped‐ cantilever structure. Let’s treat the moment reaction at A as the redundant reaction and use method of consistent deformations (also known as compatibility method or flexibility method) to solve the system. + We can use the virtual force method to find and . That is, we first apply a tracer virtual unit moment A,15 A,MA at A and then convolve the resulting virtual bending moment distribution with the curvature distributions in the two simply supported beams loaded with real external force 15 kN and support reaction M A , respectively, to find the corresponding internal virtual strain energy results. Equating these to the respective external virtual work in each case we can find and . A,15 A,MA CE474 – Structural Analysis II Equating external virtual work to internal virtual strain energy Now requiring the compatibility condition that net slope change at A should be zero, we can find M A . Now that we have found the moment distribution in the propped cantilever when it loaded by 15 kN downward force at midspan point B, we can now find vertical displacement at B, B . We will do so using virtual force approach –aside: this method is also known as “dummy load” method or “unit load” method. We will find B using three different “virtual systems”.
    [Show full text]