Table Of Content51
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
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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.
