51 TEXTS AND READINGS IN MATHEMATICS Jacobi's Lectures On Dynamics Texts and Readings in Mathematics Advisory Editor C. S. Seshadri, Chennai Mathematical Institute, Chennai. Managing Editor Rajendra Bhatia, Indian Statistical Institute, New Delhi. Editors R. B. Bapat, Indian Statistical Institute, New Delhi. V. S. Borkar, Tata Inst. of Fundamental Research, Mumbai. Probai Chaudhuri, Indian Statistical Institute, Kolkata. V. S. Sunder, Inst. of Mathematical Sciences, Chennai. M. Vanninathan, TIFR Centre, Bangalore. Jacobi's Lectures on Dynamics Second Revised Edition Delivered at the University of Königsberg in the winter semester 1842-1843 and according to the notes prepared by C. W. Brockardt Edited by A. Clebsch Translated from the original German by K. Balagangadharan Translation edited by Biswarup Banerjee U UU UB OOK AGENCY Published in lndia by Hindustan Book Agency (India) P 19 Green Park Extension New Delhi 110 016 India email: [email protected] Http://www.hindbook.com Copyright © 2009, Hindustan Book Agency (India) No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, incIuding photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner, who has also the sole right to grant licences for translation into other languages and publication thereof. All export rights for this edition vest excIusively with Hindustan Book Agency (India) . Unauthorized export is a violation of Copyright Law and is subject to legal action. ISBN 978-81-85931-91-3 ISBN 978-93-86279-62-0 (eBook) DOI 10.1007/978-93-86279-62-0 Contents Foreword .... ix 1 Introduction 1 2 The Differential Equations of Motion 7 3 Conservation of Motion of Cent re of Gravity 16 4 The Principle of Conservation of 'vis viva' 20 5 Conservation of Surface Area 33 6 The Principle of Least Action 46 7 Further considerations on the principle of least action-The Lagrange multipliers ...................... 55 8 Hamilton's Integral and Lagrange's Second Form of Dynamical Equations . . . . . . . . . . . . . 62 9 Hamilton's Form of the Equations of Motion 72 10 The Principle of the Last Multiplier . . . . . 77 11 Survey of those properties of determinants that are used in the theory of the last multiplier . . . . . . . . . . . . . . .. 92 12 The multiplier for systems of differential equations with an arbitrary number of variables ................. 98 13 Functional Determinants. Their application in setting up the Partial Differential Equation for the Multiplier. . . . . .. 109 14 The Second Form of the Equation Defining the Multiplier. The Multipliers of Step Wise Reduced Differential Equations. The Multiplier by the Use of Particular Integrals ...... 116 Lectures in Dynamics vi 15 The Multiplier for Systems of Differential Equations with Higher Differential Coefficients. Applications to a System of Mass Points Without Constraints .. .... 129 16 Examples of the Search for Multipliers. Attraction of a Point by a Fixed Cent re in a Resisting Medium and in Empty Space . . . 137 17 The Multiplier of the Equations of Motion of a System Under Constraint in the first Langrange Form . . 144 18 The Multiplier for the Equations of Motion of a Constrained System in Hamiltonian Form . . 154 19 Hamilton's Partial Differential Equation and its Extension to the Isoperimetric Problem 157 20 Proof that the integral equations derived from a complete solution of Hamilton's partial differential equation actually satisfy the system of ordinary differential equations. Hamilton's equation for free motion . 171 21 Investigation of the case in which t does not occur explicitly . .. . . 177 22 Lagrange's method of integration of first order partial differential equations in two independent variables. Application to problems of mechanics which depend only on two defining parameters. The free motion of a point on a plane and the shortest line on a surface . . 183 23 The reduction of the partial differential equation for those problems in which the principle of conservation of cent re of gravity holds . .. .. 193 24 Motion of a planet around the sun - Solution in polar coordinates . . . 200 25 Solution of the same problem by introducing the distances of the planet from two fixed points 208 26 Elliptic Coordinates . 219 Lectures in Dynamics Vll 27 Geometrie significance of elliptie coordinates on the plane and in space. Quadrature of the surface of an ellipsoid. Rectification of its lines of curvature ............ 229 28 The shortest line on the tri-axial ellipsoid. The problem of map projection .............. . 235 29 Attraction of a point by two fixed centres 247 30 Abel's Theorem .............. . 259 31 General investigations of the partial differential equations of the first order. Different forms of the integrability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 267 32 Direct proof of the most general form of the integrability condition. Introduction of the function H, which set equal to an arbitrary constant determines the p as functions of the q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 279 33 On the simultaneous solutions of two linear partial differential equations ............................ 288 34 Application of the preceding investigation to the integration of partial differential equations of the first order, and in partieular, to the case of mechanics. The theorem on the third integral derived from two given integrals of differential equations of dynamics ..................... 298 35 The two classes of integrals whieh one obtains according to Hamilton's method for problems of mechanies. Determination of the value of (ep, 'l/J) for them 307 36 Perturbation theory . 316 Supplement . . . . . . . . 328 Foreword The present supplement to C.G.J. Jacobi's collected works contains the second revised edition of the "Lectures on Dynamics" edited by A. Cleb- sch in 1866 without the five treatises from Jacobi's literary estate added to them at that time. According to the plans drawn up for the publica- tion of Jacobi's collected works the latter along with the major treatise "Nova methodus aequtionen differentiales partiellen primi ordinus inter numerum variabilium quemcunque propositas integrandi" , (New meth- ods for the integration of first order partial differential equations of any number of variables), also edited by Clebsch, and a few other shorter works will form the contents of the fifth volume. As has been remarked in the preface to the first edition of the "Lec- tures", they are based on the not es prepared with great care nad accu- racy by C. W. Borckhardt who attended the lectures given by Jacobi at the University of Königsberg in the winter semester of 1842-43. The changes made by Clebsch in the edition of Brochardt's text are minor. Also Mr. E. Lottner, the publisher of the new edition, has only made slight stylistic changes in certain places where the express ions were not precise or sufficiently clear, and for the rest has confined himself to cor- recting a few printing and computational errors remaining in the first edition. 15 March 1884 Weierstrass (Translated by Balgangadharan. Revised by B.Banerjee) Note (B. Banerjee) We have translated some of Jacobi's expressions as they were in his time to retain the fiavoilr of the original. They are: 1. vis viva(lebendige Kraft) stands for twice the kinetic energy. Lectures in Dynamics x 2. Force function (Kraftfunction) stands for potential or potential energy. 3. The principle of conservation of surface area (Das Princip der Er- haltung der Flächenräume ) stands for the principle of conservation of angular moment um.