History of Virtual Work Laws: A History of Mechanics Prospective (Science Networks. Historical Studies, Vol. 42) (Science Networks. Historical Studies (42))

This book presents a history of classical mechanics, documenting methods of study from Greece in the fourth century BC to late nineteenth-century Europe. The text assesses effectiveness of forces based on the virtual motion of their points of application. [Duhem](http://www.encyclopedia.com/topic/Pierre_Maurice_Marie_Duhem.aspx#1) mentioned toward the end Title Page 3 Copyright Page 4 Preface 5 Table of Contents 7 1 Introduction 13 1.1 Virtual velocity laws 14 1.2 Virtual displacement laws 16 1.3 Virtual work laws as principles of mechanics 17 1.4 Virtual work laws as theorems ofmechanics 21 1.5 Contemporary tendencies 22 1.6 Final remarks. The rational justification of virtual work laws 24 2 Logic status of virtual work laws 26 2.1 The theorem of virtual work 28 2.1.1 Proofs of the virtual work theorems in the literature 34 2.1.1.1 Physics and rational mechanics treatises 34 2.1.1.2 Statics handbooks 35 2.1.1.3 Poinsot’s proof 37 2.2 The principle of virtual work 38 2.2.1 Force as a primitive concept 39 2.2.1.1 Equilibrium case 39 2.2.1.2 Motion case 40 2.2.2 Work as a primitive concept 42 2.2.2.1 Equilibrium case 42 2.2.2.2 Motion case 43 3 Greek origins 44 3.1 Different approaches to the law of the lever 45 3.1.1 Aristotelian mechanics 45 3.1.1.1 Physica and De caelo 46 3.1.1.2 Mechanica problemata 49 3.1.1.3 A law of virtual work 54 3.1.2 Archimedean mechanics 56 3.1.2.1 Proof of the law of the lever 59 3.2 Themechanics of Hero of Alexandria 62 3.2.1 The principles of Hero’s mechanics 64 3.2.1.1 A law of virtual work 66 3.2.1.2 Hero’s inclined plane law 69 3.3 The mechanics of Pappus of Alexandria 70 3.3.1 Pappus’ inclined plane law 71 4 Arabic and Latin science of weights 73 4.1 Arabicmechanics 76 4.1.1 Weight as an active factor in Arabic mechanics 78 4.1.1.1 Liber karastonis 79 4.1.1.2 Kitab al-Qarastun 83 4.1.2 Comments on the Arabic virtual work law 84 4.2 Latin mechanics 85 4.2.1 Weight as a passive factor in the Latin mechanics 90 4.2.2 Propositions 90 4.2.2.1 Proposition I 91 4.2.2.2 Proposition II 94 4.2.2.3 Proposition VI. The law of the Lever 96 4.2.2.4 Proposition VIII 96 4.2.2.5 Proposition X. The law of the inclined plane 98 4.2.3 Comments on the Latin virtual work law 99 5 Italian Renaissance statics 100 5.1 Renaissance engineering 104 5.1.1 Daniele Barbaro and Buonaiuto Lorini 105 5.2 Nicolò Tartaglia 106 5.2.1 Definitions and petitions 107 5.2.2 Propositions 107 5.2.2.1 Proof of propositions I–IV 109 5.2.2.2 The law of the lever 110 5.2.2.3 The law of the inclined plane 112 5.3 Girolamo Cardano 113 5.3.1 De subtitilate 114 5.3.2 De opus novum 116 5.4 Guidobaldo dal Monte 117 5.4.1 The centre of gravity 118 5.4.2 The balance 118 5.4.3 The virtual work law 124 5.5 Giovanni Battista Benedetti 125 5.5.1 Effect of the position of a weight on its heaviness 125 5.5.2 Errors of Tartaglia and Jordanus 127 5.6 Galileo Galilei 129 5.6.1 The concept of moment. A law of virtual velocities 130 5.6.2 A law of virtual displacements 136 5.6.3 Proof of the law of the inclined plane 140 6 Torricelli’s principle 143 6.1 The centrobaric 143 6.2 Galileo’s centrobaric 146 6.3 Torricelli’s joined heavy bodies 148 6.3.1 Torricelli’s fundamental concepts on the centre of gravity 149 6.4 Torricelli’s principle 152 6.4.1 Analysis of the aggregate of two bodies 154 6.4.2 Torricelli’s principle as a criterion of equilibrium 156 6.5 Evolution of Torricelli’s principle. Its role in virtual work laws 161 6.5.1 A restricted form of Torricelli’s principle 162 7 European statics during the XVI and XVII centuries 164 7.1 French statics 164 7.1.1 Gille Personne de Roberval 167 7.1.1.1 The inclined plane law 167 7.1.1.2 The rule of the parallelogram 168 7.1.2 René Descartes 171 7.1.2.1 The concept of force 171 7.1.2.2 The application to simple machines 174 7.1.2.3 The refusal of virtual velocities 177 7.1.2.4 Displacements at the very beginning of motion 178 7.1.2.5 A possible precursor 180 7.1.3 Blaise Pascal 182 7.1.4 Post Cartesians 183 7.2 Nederland statics 184 7.2.1 Simon Stevin 185 7.2.1.1 The rule of the parallelogram of forces 187 7.2.1.2 The law of virtual work 191 7.2.2 Christiaan Huygens 194 7.3 British statics 196 7.3.1 John Wallis 197 7.3.2 Isaac Nevton 200 8 The principle of virtual velocities 202 8.1 The concept of force in the XVIII century 202 8.1.1 Newtonian concept of force 202 8.1.2 Leibnizian concept of force 204 8.2 Johann Bernoulli mechanics 206 8.2.1 Dead and living forces according to Bernoulli 206 8.2.2 The rule of energies 208 8.3 Varignon: the rule of energies and the law of composition of forces 217 8.3.1 Elements of Varignon’s mechanics 217 8.3.2 The rule of the parallelogram versus the rule of energies 220 9 The Jesuit school of the XVIII century 223 9.1 Vincenzo Angiulli and Vincenzo Riccati 224 9.1.1 The principle of actions of Vincenzo Angiulli 224 9.1.1.1 The action of a force 225 9.1.1.2 The principle of actions 227 9.1.1.3 The measure of actions 229 9.1.1.4 The principle of action and the principles of statics 231 9.1.1.5 The applications to simple machines 234 9.1.2 The principle of actions of Vincenzo Riccati 236 9.2 Ruggiero Giuseppe Boscovich 239 9.2.1 A virtual work law for Saint Peter’s dome 240 9.2.1.1 The mechanism of failure and the forces 241 10 Lagrange’s contribution 243 10.1 First introduction of the virtual velocity principle 246 10.1.1 The first ideas about a new principle of mechanics 246 10.1.2 Recherches sur la libration de la Lune 248 10.1.2.1 Setting of the astronomical problem 251 10.1.2.2 The symbolic equation of dynamics 253 10.1.2.3 The virtual velocity principle 256 10.1.3 The Théorie de la libration de la Lune 257 10.2 Méchanique analitique and Mécanique analytique 258 10.2.1 Méchanique analitique 259 10.2.1.1 Constraint reactions 264 10.2.2 Mécanique analytique 265 10.2.2.1 Criticisms of Lagrange’s proof 269 10.3 The Théorie des fonctions analytiques 270 10.4 Generalizations of the virtual velocity principle to dynamics 274 10.4.1 The calculus of variations 279 10.4.2 Elements of D’Alembert’s mechanics 280 10.4.2.1 D’Alembert principle 283 11 Lazare Carnot’s mechanics of collision 286 11.1 Carnot’s laws of mechanics 290 11.1.1 The first fundamental equation of mechanics 292 11.1.2 Geometric motions 294 11.1.3 The second fundamental equation of mechanics 296 11.2 Gradual changing of motion. A law of virtual work 298 11.3 The moment of activity 300 12 The debate in Italy 303 12.1 The criticisms on the evidence of the principle 304 12.1.1 Vittorio Fossombroni 304 12.1.1.1 Invariable distance systems 305 12.1.1.2 The equation of forces 306 12.1.1.3 The equation of moments 308 12.1.2 Girolamo Saladini 310 12.1.3 François Joseph Servois 312 12.2 The criticisms on the use of infinitesimals 315 12.2.1 Gabrio Piola 316 12.2.1.1 Piola’s principles of material point mechanics 316 12.2.1.2 System of free material points 318 12.2.1.3 System of constrained material points 319 13 The debate at the École polytechnique 320 13.1 One of the first professor of mechanics, Gaspard de Prony 322 13.1.1 Proof from the composition of forces rule 323 13.2 Joseph Fourier 324 13.2.1 First proof 326 13.2.2 Second proof 328 13.2.3 Third proof 329 13.3 André Marie Ampère 331 13.4 Pierre Simon Laplace 335 14 Poinsot’s criticism 338 14.1 Considérations sur le principe des vitesses virtuelles 339 14.2 Théorie générale de l’équilibre et du mouvement des systèmes 342 14.2.1 Poinsot’s principles of mechanics 345 14.2.1.1 System of material points constrained by a unique equation 347 14.2.1.2 System of material points constrained by more equations 349 14.3 Demonstration of the virtual velocity principle 351 15 Complementary virtual work laws 355 15.1 Augustin Cauchy formulation 356 15.1.1 Kinematics of plane rigid bodies 358 16 The treatises of mechanics 363 16.1 Siméon Denis Poisson 364 16.2 Jean Marie Duhamel 367 16.3 Gaspard Gustave Coriolis 369 17 Virtual work laws and continuum mechanics 376 17.1 First applications 376 17.1.1 Joseph Louis Lagrange 376 17.1.1.1 Mono-dimensional continuum 377 17.1.1.2 Three-dimensional continuum 378 17.1.2 Navier’s equations of motion 382 17.2 Applications in the theory of elasticity 384 17.2.1 Alfred Clebsch 384 17.3 The Italian school 388 17.3.1 Gabrio Piola 389 17.3.2 Eugenio Beltrami 391 17.3.3 Enrico Betti 393 18 Thermodynamical approach 395 18.1 Pierre Duhem’s concept of oeuvre 396 18.1.1 Virtual transformations 397 18.1.2 Activity, energy and work 398 18.1.3 Rational mechanics 401 18.1.3.1 Free systems 401 18.1.3.2 Constrained systems 402 Appendix. Quotations 404 A.1 Chapter 1 404 A.2 Chapter 2 405 A.3 Chapter 3 406 A.4 Chapter 4 408 A.5 Chapter 5 411 A.6 Chapter 6 422 A.7 Chapter 7 425 A.8 Chapter 8 432 A.9 Chapter 9 436 A.10 Chapter 10 440 A.11 Chapter 11 447 A.12 Chapter 12 451 A.13 Chapter 13 453 A.14 Chapter 14 456 A.15 Chapter 15 462 A.16 Chapter 16 463 A.17 Chapter 17 466 A.18 Chapter 18 470 References 472 Index 487 Annotation The book presents a history of classical mechanics by focusing on issues of equilibrium. The historical point of view adopted here restricts attention to cases where the effectiveness of forces is assessed on the basis of the virtual motion of their points of application. For completeness, hints of the alternative approach are also referred, the Archimedean for ancient mechanics and the Newtonian for modern mechanics. The laws resulting from consideration of virtual motions are named laws of virtual work. The modern formulations of the principle of virtual work are only a particular form of them. The book begins with the first documented formulations of laws of virtual work in the IV century BC in Greece and proceeds to the end of the XIX century AD in Europe. A significant space is devoted to Arabic and Latin mechanics of Middle Ages. With the Renaissance it began to appear slightly different wordings of the laws, which were often proposed as unique principles of statics. The process reached its apex with Bernoulli and Lagrange in the XVIII century. The book ends with some chapters dealing with the discussions that took place in the French school on the role of the Lagrangian version of the law of virtual work and its applications to continuum mechanics