A TREATISE ON DYNAMICS OF A PARTICLE A TEEATISE ON DYNAMICS OF A PARTICLE WITH NUMEROUS EXAMPLES BY EDWARD JOHN ROUTH, Sc. D., LL. D., M. A., F. RS., c. HON. FELLOJW OF PETERHOUSE, CAMBRIDGE FELLOW OF THE UNIVERSITY OF LONDON G. E. STECHERT CO. NEW YORK yi PREFACE. by which the whole solution of a dynamical problem can be made to depend on a single integral. The last word has not yet been said on these problems. The student finds as he proceeds much left to discover and many new questions to ask. When we extend our studies so as to include the planetary perturbations and to take account of the finite size of the bodies the mathematical difficulties are much increased. In the dynamics of a particle we confine ourselves to simpler problems and easier mathematics. As the subject of dynamics is usually read early in the mathematical course, the student cannot be expected to master all its difficulties at once. In this treatise the parts intended for a first reading are printed in large type and the student is advised to pass over the other parts until they are referred to later on. The same problem may be attacked on many sides and we therefore have several different ways of finding a solution. In what follows the most elementary method has in general been put first, other solutions being given later on. For the sake of simplicity they have also generally been treated first in two dimensions. In these ways the difficulties of dynamics are separated from those of pure geometry and it is hoped that both difficulties may thus be more easily overcome. Some of the examples have been fully worked out, on others hints have been given. Many of these have been selected from the Tripos and College papers in order that they may the better indicate the recent directions of dynamical thought. I cannot conclude without thanking Mr Dickson of Peterhouse. He has kindly assisted me in correcting most of the proofs and has given material aid by his verifications and suggestions. EDWARD J-. ROUTEL PETERHOUSE, 1898. CONTENTS. CHAPTER I. ELEMENTARY CONSIDERATIONS. ABTS. 1 30. Velocity and acceleration . 31 38. Cartesian, polar and intrinsic components 39 40. Belative motion 41 45. Angular velocity . 46 48. Units of space and time 49 62. Laws of motion 63 67. Units of mass and force 68 72. Vis viva and work 73 79. The two solutions of the equations of motion . 80 91. Impulsive forces and impacts . 92 93. Motion of the centre of gravity 94. Examples on impacts PAGES 110 1113 1415 1516 1617 1723 2326 2627 2834 3541 4142 4246 CHAPTER II. RECTILINEAR MOTION. 95 103. Solution of the equation. Ambiguities .... 47 53 104 117. A heavy particle in vacuo and in a resisting medium. Bough chords 53 60 118 122. Linear equation and harmonic motion .... 60 64 Vlll CONTENTS. ARTS. PAGES 123 136. Centre of force. Discontinuity of friction, of resistance and of central forces ....... 64 73 137 142. Small oscillations. Magnification ..... 73 76 143 147. Chords of quickest descent. Smooth. Kough . . 77 79 148 150. Infinitesimal impulses 80 82 151 153. Theory of dimensions ....... 82 83 CHAPTER III. MOTION OF PROJECTILES. 154 161. Parabolic motion 84 92 162 167. Eesistance varies as the velocity 92 95 168 180. Other laws of resistance. Special cases .... 95 102 CHAPTER IV. CONSTBAINED MOTION IN TWO DIMENSIONS. 181 190. The two resolutions. Work function .... 103 110 191 192. Kough curves , HO 112 193 196. Zero pressure 112 115 197 198. Moving curves of constraint 115 117 199 203. Time of describing an arc. Subject of integration infinite 117 120 204 212. Motion in a cycloid. Smooth rough and a resisting medium 120 126 213 220. Motion in a circle. Coaxial circles G 126 133 CHAPTER V. MOTION IN TWO DIMENSIONS. 22133. Moving axes. Eelative motion 134141 234 245. DAlemberts principle fec. 141 147 246 254. Vis viva and energy 147 154 CONTENTS. ABTS. 255 256. Botatmg field of force. Jacobi . 257 258. Eelative vis viva. Coriolis 259 267. Moments and resolutions .... 268270... --This text refers to an alternate Paperback edition.