Mechanistic Images in Geometric Form : Heinrich Hertz's 'Principles of Mechanics'

This book gives an analysis of Hertz's posthumously published 'Principles of Mechanics' in its philosophical, physical and mathematical context. In a period of heated debates about the true foundation of physical sciences, Hertz's book was conceived and highly regarded as an original and rigorous foundation for a mechanistic research program. Insisting that a law-like account of nature would require hypothetical unobservables, Hertz viewed physical theories as (mental) images of the world rather than the true design behind the phenomena. This paved the way for the modern conception of a model. Rejecting the concept of force as a coherent basic notion of physics he built his mechanics on hidden masses (the ether) and rigid connections, and formulated it as a new differential geometric language. Recently many philosophers have studied Hertz's images and historians of physics have discussed his forceless mechanics. The present book shows how these aspects, as well as the hitherto overlooked mathematical aspect, form an integrated whole research on electromagnetism. Therefore it is also a case study of the strong interactions between philosophy, physics and mathematics. Moreover, the book presents an analysis of the genesis of many of the central elements of Hertz's mechanics based on his manuscripts and drafts. Hertz's research programs was cut short by the advent of relativity theory but it's image theory influenced many philosophers as well as some physicists and mathematicians and its geometric form had a lasting influence on advanced expositions of mechanics. Contents......Page 10 1.1 Aim and structure of the book......Page 16 1.2 General outline of Hertz’s Mechanics......Page 17 1.3 What is new in this book?......Page 20 1.4 The publication of Hertz’s book and the preserved drafts......Page 21 1.5 Notation, quotes, references, etc......Page 22 2 The principles of mechanics before Hertz......Page 23 2.1 Principles and laws......Page 24 2.2 Foundations of mechanics......Page 35 2.3 Basic notions......Page 37 2.4 Novel expositions and critical works......Page 42 3 Mechanization of physics......Page 45 3.1 The decline of the mechanistic world view......Page 53 4.1 Forces and atoms......Page 55 4.2 The problematization of distance forces. Field theory......Page 56 4.3 Rejections of atomism......Page 60 4.4 Energetics......Page 62 5.1 Childhood and student years (1857–1883)......Page 65 5.2 Privat Dozent in Kiel (1883–1885)......Page 69 5.3 Professor in Karlsruhe (1885–1889)......Page 71 5.4 Professor in Bonn (1889–1894)......Page 72 6.1.1 Axiomatization......Page 78 6.1.2 Mechanization......Page 81 6.1.3 The elimination of distance forces......Page 82 6.2 Research on gravitation......Page 86 6.3 Ether......Page 88 6.4 An energetic beginning......Page 92 6.5 Chronology of the drafts......Page 95 7 Images of nature......Page 98 7.1 A comparison of Hertz’s and Helmholtz’s signs and images......Page 100 7.2 Correctness......Page 101 7.3 Logical permissiblity......Page 102 7.4 Appropriateness......Page 104 7.4.1 Distinctness......Page 105 7.4.2 Simplicity......Page 107 7.5 The relation among the criteria......Page 108 8.1 Images in the Kiel Lectures......Page 112 8.2 The parable of the paper money......Page 114 8.4 Comparison of the 1894 images with earlier concepts......Page 116 8.5 Concepts in the Mechanics related to images......Page 121 9.1 Principles of mechanics......Page 126 9.2 The Newtonian–Laplacian image......Page 127 9.3 The energetic image......Page 129 9.4 Hertz’s image......Page 130 9.5 Conclusion of the comparison......Page 132 10.1 Scientific representations......Page 134 10.2 A Kantian division......Page 136 10.3 A metaphysics of corporeal nature......Page 138 10.4 Kantianism in the first draft of Hertz’s Mechanics. Existence problems......Page 139 10.5 The division between kinematics and dynamics......Page 140 11.1 Space......Page 142 11.1.1 Pure geometry......Page 143 11.1.2 Applied geometry......Page 146 11.2 Time......Page 148 11.3 Mass. The constitution of matter......Page 149 11.3.1 The constitution of matter. Book one......Page 150 11.3.2 The constitution of matter. Book two......Page 156 12.1 Hertz’s line element......Page 161 12.2 Vanishing identical material points rejected......Page 164 12.3 Massenteilchen appear......Page 166 12.4 A matter of space......Page 169 12.5 The Massenteilchen shrink and become matter free......Page 171 12.6 Conclusion......Page 173 13.2 Why the geometric form?......Page 174 13.3 Direction, angle, and curvature in the printed book......Page 178 13.4 Direction, angle, and curvature in Hertz’s manuscripts......Page 180 14.2 Components and reduced components of a displacement......Page 188 14.3 Vector quantities......Page 191 14.4 Kinematic concepts......Page 193 14.5 Vector quantities in Hertz’s drafts......Page 195 14.6 The (reduced) components in Hertz’s drafts......Page 196 14.7 The origin of Hertz’s tensor analysis......Page 198 14.8 Interaction between physical content and mathematical form......Page 200 15.1 Connections rather than forces......Page 202 15.2 Derivation of the equations of connection......Page 204 15.3 Holonomic and non-holonomic systems......Page 207 16 The fundamental law......Page 213 17.1 Straightest paths......Page 217 17.2 Dynamics of free systems......Page 219 18.1 Routh and modified Lagrangians......Page 223 18.2 Hidden cyclic motion. J.J. Thomson......Page 225 18.3 A simple standard example......Page 226 18.4 Helmholtz on adiabatic cyclic systems......Page 228 18.6 R. Liouville: One cyclic coordinate suffice......Page 232 19.1 Guided systems......Page 234 19.2 Systems acted on by forces......Page 235 20.1 Adiabatic cyclic systems......Page 240 20.2 Conservative systems......Page 242 20.3 Hidden non-holonomic connections are not allowed......Page 244 20.4 The approximative character of cyclic and conservative systems......Page 245 21 Integral principles......Page 250 21.1 Shortest and geodesic paths......Page 251 21.2 Integral principles of mechanics......Page 252 22.1 Hölder’s rescue of Hamilton’s principle......Page 255 22.2 Repeated independent mistakes, rejections and rescues......Page 258 23.1 The straightest distance......Page 262 23.2 The characteristic and principal functions......Page 264 24.1 Gauss and Hamilton on geodesics, optics and dynamics......Page 267 24.2 Liouville and Lipschitz on the principle of least action......Page 270 24.3 Trajectories as geodesics......Page 273 24.4 Hertz and the mathematicians......Page 276 25.1 Practical applications......Page 278 25.2 Validity and applicability of the fundamental law......Page 279 25.3 Constructability of forces......Page 280 25.4 Vitalism, teleology, reductionism, and mechanism in nineteenth-century biology......Page 282 25.5 Hertz on living systems......Page 283 25.6 ‘Permissible,’ ‘probable’?......Page 287 25.7 Applicability and correctness......Page 288 26 Force-producing models......Page 289 27.1 Reception......Page 293 27.2 Extensions and applications......Page 299 27.3 Impact......Page 301 28 List of conclusions......Page 305 Appendix......Page 310 Bibliography......Page 314 C......Page 328 E......Page 329 I......Page 330 M......Page 331 P......Page 332 W......Page 333 Contents 10 1 Introduction 16 1.1 Aim and structure of the book 16 1.2 General outline of Hertz’s Mechanics 17 1.3 What is new in this book? 20 1.4 The publication of Hertz’s book and the preserved drafts 21 1.5 Notation, quotes, references, etc 22 2 The principles of mechanics before Hertz 23 2.1 Principles and laws 24 2.2 Foundations of mechanics 35 2.3 Basic notions 37 2.4 Novel expositions and critical works 42 3 Mechanization of physics 45 3.1 The decline of the mechanistic world view 53 4 Problematization of the concept of force 55 4.1 Forces and atoms 55 4.2 The problematization of distance forces. Field theory 56 4.3 Rejections of atomism 60 4.4 Energetics 62 5 A biographical survey 65 5.1 Childhood and student years (1857–1883) 65 5.2 Privat Dozent in Kiel (1883–1885) 69 5.3 Professor in Karlsruhe (1885–1889) 71 5.4 Professor in Bonn (1889–1894) 72 6 Hertz’s road to mechanics 78 6.1 Hertz’s electromagnetic work as a background for his mechanics 78 6.1.1 Axiomatization 78 6.1.2 Mechanization 81 6.1.3 The elimination of distance forces 82 6.2 Research on gravitation 86 6.3 Ether 88 6.4 An energetic beginning 92 6.5 Chronology of the drafts 95 7 Images of nature 98 7.1 A comparison of Hertz’s and Helmholtz’s signs and images 100 7.2 Correctness 101 7.3 Logical permissiblity 102 7.4 Appropriateness 104 7.4.1 Distinctness 105 7.4.2 Simplicity 107 7.5 The relation among the criteria 108 8 Hertz’s earlier ideas about images 112 8.1 Images in the Kiel Lectures 112 8.2 The parable of the paper money 114 8.3 The colorless theory and the gay garment 116 8.4 Comparison of the 1894 images with earlier concepts 116 8.5 Concepts in the Mechanics related to images 121 9 Images of mechanics 126 9.1 Principles of mechanics 126 9.2 The Newtonian–Laplacian image 127 9.3 The energetic image 129 9.4 Hertz’s image 130 9.5 Conclusion of the comparison 132 10 Kantianism. A-priori and empirical elements of images 134 10.1 Scientific representations 134 10.2 A Kantian division 136 10.3 A metaphysics of corporeal nature 138 10.4 Kantianism in the first draft of Hertz’s Mechanics. Existence problems 139 10.5 The division between kinematics and dynamics 140 11 Time, space and mass 142 11.1 Space 142 11.1.1 Pure geometry 143 11.1.2 Applied geometry 146 11.2 Time 148 11.3 Mass. The constitution of matter 149 11.3.1 The constitution of matter. Book one 150 11.3.2 The constitution of matter. Book two 156 12 The line element: The origin of the Massenteilchen 161 12.1 Hertz’s line element 161 12.2 Vanishing identical material points rejected 164 12.3 Massenteilchen appear 166 12.4 A matter of space 169 12.5 The Massenteilchen shrink and become matter free 171 12.6 Conclusion 173 13 Hertz’s geometry of systems of points 174 13.1 Geometrization of mechanics 174 13.2 Why the geometric form? 174 13.3 Direction, angle, and curvature in the printed book 178 13.4 Direction, angle, and curvature in Hertz’s manuscripts 180 14 Vector quantities and their components 188 14.1 Introduction 188 14.2 Components and reduced components of a displacement 188 14.3 Vector quantities 191 14.4 Kinematic concepts 193 14.5 Vector quantities in Hertz’s drafts 195 14.6 The (reduced) components in Hertz’s drafts 196 14.7 The origin of Hertz’s tensor analysis 198 14.8 Interaction between physical content and mathematical form 200 15 Connections. Material systems 202 15.1 Connections rather than forces 202 15.2 Derivation of the equations of connection 204 15.3 Holonomic and non-holonomic systems 207 16 The fundamental law 213 17 Free systems 217 17.1 Straightest paths 217 17.2 Dynamics of free systems 219 18 Cyclic coordinates 223 18.1 Routh and modified Lagrangians 223 18.2 Hidden cyclic motion. J.J. Thomson 225 18.3 A simple standard example 226 18.4 Helmholtz on adiabatic cyclic systems 228 18.5 What is new in Hertz’s Mechanics? 232 18.6 R. Liouville: One cyclic coordinate suffice 232 19 Unfree systems. Forces 234 19.1 Guided systems 234 19.2 Systems acted on by forces 235 20 Cyclic and conservative systems 240 20.1 Adiabatic cyclic systems 240 20.2 Conservative systems 242 20.3 Hidden non-holonomic connections are not allowed 244 20.4 The approximative character of cyclic and conservative systems 245 21 Integral principles 250 21.1 Shortest and geodesic paths 251 21.2 Integral principles of mechanics 252 22 A history of non-holonomic constraints 255 22.1 Hölder’s rescue of Hamilton’s principle 255 22.2 Repeated independent mistakes, rejections and rescues 258 23 Hertz on the Hamilton formalism 262 23.1 The straightest distance 262 23.2 The characteristic and principal functions 264 24 Mathematicians on the geometrization of the Hamilton–Jacobi formalism 267 24.1 Gauss and Hamilton on geodesics, optics and dynamics 267 24.2 Liouville and Lipschitz on the principle of least action 270 24.3 Trajectories as geodesics 273 24.4 Hertz and the mathematicians 276 25 Hertz on the domain of applicability of his mechanics 278 25.1 Practical applications 278 25.2 Validity and applicability of the fundamental law 279 25.3 Constructability of forces 280 25.4 Vitalism, teleology, reductionism, and mechanism in nineteenth-century biology 282 25.5 Hertz on living systems 283 25.6 ‘Permissible,’ ‘probable’? 287 25.7 Applicability and correctness 288 26 Force-producing models 289 27 Reception, extension and impact 293 27.1 Reception 293 27.2 Extensions and applications 299 27.3 Impact 301 28 List of conclusions 305 Appendix 310 Bibliography 314 Index 328 A 328 B 328 C 328 D 329 E 329 F 330 G 330 H 330 I 330 J 331 K 331 L 331 M 331 N 332 O 332 P 332 R 333 S 333 T 333 U 333 V 333 W 333 This book gives an analysis of Hertz's posthumously published Principles of Mechanics in its philosophical, physical and mathematical context. In a period of heated debates about the true foundation of physical sciences, Hertz's book was conceived and highly regarded as an original and rigorous foundation for a mechanistic research program. Insisting that a law-like account of nature would require hypothetical unobservables, Hertz viewed physical theories as (mental) images of the world rather than the true design behind the phenomena. This paved the way for the modern conception of a model. Rejecting the concept of force as a coherent basic notion of physics he built his mechanics on hidden masses (the ether) and rigid connections, and formulated it as a new differential geometric language. Recently many philosophers have studied Hertz's image theory and historians of physics have discussed his forceless mechanics. The present book shows how these aspects, as well as the hitherto overlooked mathematical aspects, form an integrated whole which is closely connected to the mechanistic world view of the time and which is a natural continuation of Hertz's earlier research on electromagnetism. Therefore it is also a case study of the strong interactions between philosophy, physics and mathematics. Moreover, the book presents an analysis of the genesis of many of the central elements of Hertz's mechanics based on his manuscripts and drafts. Hertz's research program was cut short by the advent of relativity theory but its image theory influenced many philosophers as well as some physicists and mathematicians and its geometric form had a lasting influence on advanced expositions of mechanics. "This book gives an analysis of Hertz's posthumously published Principles of Mechanics in its philosophical, physical and mathematical context. Recently many philosophers have studied Hertz's image theory and historians of physics have discussed his forceless mechanics. The present book shows how these aspects, as well as the hitherto overlooked mathematical aspects, form an integrated whole which is closely connected to the mechanistic world view of the time and which is a natural continuation of Hertz's earlier research on electromagnetism. Thereby it is a case study in the strong interactions between philosophy, physics and mathematics. Moreover, the book presents an analysis of the genesis of many of the central elements of Hertz's mechanics based on his manuscripts and drafts."--BOOK JACKET This book presents an analysis of Heinrich Hertz's posthumously published 'Principles of Mechanics' in its philosophical, physical and mathematical context. It is the first book to comprehensively examine Hertz's important ideas on mechanics and the author draws on the original manuscripts to reveal the creative process of Hertz throughout the writing of his book. - ;This book gives an analysis of Hertz's posthumously published Principles of Mechanics in its philosophical, physical and mathematical context. In a period of heated debates about the true foundation of physical sciences, Hertz's b This book presents an analysis of Heinrich Hertz's posthumously published 'Principles of Mechanics' in its philosophical physical and mathematical context. It comprehensively examines Hertz's ideas on mechanics and the author draws on the original manuscripts to reveal the creative process of Hertz.