The Theory of the Top Volume II

Transcription

1

2

3 Felix Klein Arnold Sommerfeld The Theory of the Top Volume II Development of the Theory in the Case of the Heavy Symmetric Top Raymond J. Nagem Guido Sandri Translators Preface to Volume I by Michael Eckert Birkhäuser Boston Basel Berlin

4 Raymond J. Nagem Boston University Boston, MA USA Guido Sandri Boston University Boston, MA USA ISBN e-isbn DOI / Springer New York Dordrecht Heidelberg London Library of Congress Control Number: Mathematics Subject Classification (2010): 01A75, 33E05, 70E05, 70E15, 70E17, 70E18, 70E40, 70E45, 70E50 Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Birkhäuser is part of Springer Science+Business Media (

5 Contents Advertisement.... ix Volume II. Development of the Theory in the Case of the Heavy Symmetric Top. Chapter IV. The general motion of the heavy symmetric top. Introduction to elliptic integrals. 1. Intuitive discussion of the expected forms of motion; preliminaryagreements Intuitive discussion of the expected forms of motion; continuation andconclusion Quantitative treatment of the general motion of the heavy symmetric top. Execution of the six required integrations General periodicity properties of the motion. Preliminaries on the behavior of the elliptic integrals for a circulation of the integration segment. Integral representation of α, β, γ, δ On the relation between the motions of different tops that yield the same impulse curve, and on the motion of the sphericaltop Confirmation of the forms of motion of the spherical top developed in the first sections; the characteristic curves of the third order in the case e =

6 vi Contents. 7. The characteristic curves of the third order for arbitrary position of the initial circle e; distinction between strong andweaktops On the numerical calculation of the elliptic integrals for t and ψ On the approximate calculation of the top trajectories Chapter V. On special forms of motion of the heavy symmetric top, particularly pseudoregular precession, and on the stability of motion. 1. Regular precession and its neighboring forms of motion Pseudoregular precession; resolution of the paradoxes of the motion of thetop Popular explanations of the phenomena of the top intheliterature On the stability of the upright top. Geometric discussion Continuation. Analytic treatment of the motion of the upright top altered by an impact. Formulas for pseudoregular precession with small precession circle Generalities on the stability and lability of motion Energy criteria for the stability of equilibrium and motion On the method of small oscillations Onthemotion oftheheavyasymmetrictop Chapter VI. Representation of the motion of the top by elliptic functions. 1. The Riemann surface (u, U) Behavior of the elliptic integrals on the Riemann surface The image of the Riemann surface (u, U)inthet-plane Representation of α, β, γ, δ by ϑ-quotients The trajectory of the apex of the top, the polhode and herpolhode curves, etc., represented by ϑ-quotients Numerical calculation of the motion by ϑ-series Representation of the motion of the force-free top by elliptic functions Conjugate Poinsot motions. Jacobi s theorem on the relation between the motion of the force-free asymmetric top andtheheavysphericaltop vi

7 Contents. vii 9. The Lagrange equations for α, β, γ, δ of the heavy spherical top and their direct integration. Relation between the motion of the spherical top and a problem in particle mechanics Appendix to Chapter VI. 10. Thetop onthehorizontal plane Addenda and Supplements Translators Notes References Index vii

8

9 AdvertisementofVolumeIIoftheTheoryoftheTop (from the notices of the B.G. T e u b n e r publishing company in Leipzig). As a continuation of the first volume of the Theory of the Top that appeared in the summer of 1897, the second volume now follows. The first volume established the general kinematic and kinetic foundations of the theory; this volume poses, above all, the exercise of discussing the motion of the symmetric top with a fixed support point, under the influence of gravity, in all its details. Some related problems the motion of the general top under the influence of gravity and the Poinsot motion of the force-free top for a general mass distribution are more considered in passing and used for comparison than treated exhaustively. The presentation is given a somewhat broader basis only for the discussion of questions concerning the stability of motion, since this currently developing theory may claim a special interest at the present time. In this part of the work, the definitions and formulations retain sufficient generality to encompass arbitrary mechanical systems. The top appears here only as a particularly instructive example, or, if one will, an idea-forming motive. Moreover, the concept of the stability of motion is conceived here in an essentially different manner than in the relevant textbooks (of T h o m s o n and T a i t or R o u t h), but naturally in a way that subsumes the generally accepted concept of the stability of equilibrium. For what concerns the actual subject of the present volume, the motion of the heavy symmetric top, the greatest possible comprehensiveness is sought in the treatment of the problem. It is therefore not sufficient to present a general formal treatment of the problem; we also seek in the sense of the principles set out in the Introduction to open the way for full geometric and mechanical understanding of the motion, which is, without question, a not less important goal for the treatment of a mechanical problem than the analytic command of the subject. On this basis, Chapter IV commences with a qualitative description of the trajectory of the apex of the top, which only later is confirmed

10 x Advertisement. through a precise quantitative discussion of the motion. The integration of the differential equations is first accomplished in a geometric manner, whereby certain known first integrals of the motion are constructed as simple properties of the impulse vector. From the same point of view, the unquestionably best analytic method for the calculation of the motion of the top, its representation by elliptic functions, is reserved until the last chapter of this volume, in favor of the representation by elliptic integrals, which indeed is less complete, but may at first lie nearer to geometric intuition and mechanical interpretation. On the other hand, it is necessary, if one can speak of a truly complete treatment of the problem, to pursue the analytic developments to an actual numerical calculation of the motion of the top. The conclusion of the fourth chapter thus presents an introduction to numerical calculation on the basis of the Legendre integral tables, as well as a method for the derivation of approximation formulas by which one can, in the practically most important cases, directly replace the exact formulas. In Chapter VI, the question of numerical calculation is taken up once again, and answered in the most satisfactory manner with the help of the ϑ-series (including the estimation of the error bound). Chapter V treats of some particular and particularly distinctive types of motion. Two such motions are emphasized, which are designated as pseudoregular precession and the upright motion of the top. Pseudoregular precession is the motion that occurs under the usual experimental conditions of a sufficiently large eigenrotation. It hardly differs, considered externally, from actual regular precession. The paradox that is associated with this motion is extensively discussed, and is reduced to an imprecision of observation. Since most popular attempts to explain the motion of the top view regular precession as the most practically important phenomenon, there follows a short summary and critique of the popular top literature. The upright motion of the top is a uniform rotation about the vertically positioned figure axis. This motion is well known as stable for a sufficiently large rotational velocity, and as labile for a smaller velocity (where the meaning of these words is still to be discussed). Among the x

11 Advertisement. xi motions that result from a disturbance of the labile state of motion, an asymptotic case occurs that is of particular importance with respect to the associated general stability considerations. In the representation of the motion through elliptic functions in Chapter VI, the fundamental meaning of the rotation parameters α, β, γ, δ appears in full light. In these parameters, the representation of the motion attains a simplicity and clarity that is not otherwise possible. Moreover, all necessary developments from the theory of elliptic functions are reproduced in the book with some completeness, so that the relevant parts of the presentation can be directly regarded as an introduction to this theory. It does not appear improper, from a didactic point of view, to attach such an introduction to a specific example. The present presentation differs from others in that the connection to the general theory is brought out in a particularly clear manner through the detailed use of geometric relations. The concluding section of this volume may be of particular interest. Here the integration problem of the motion of the top is taken up once more, and indeed on the basis of the general Lagrange equations with α, β, γ, δ themselves as the coordinates. It is shown that these equations are the so-called Hermite Lamé differential equations, and that their integrals can be directly written in the form of elliptic functions without any intermediate calculations worth mentioning. At the same time, there follows from the form of these equations the remarkable fact that the motion of the top can be identified with the motion of a spherical pendulum in a space of four dimensions. After the pure theory of the motion of the top has thus been brought to a certain conclusion, it will be shown in the third and final volume of the book to what extent this theory coincides with experience, or what modifications must be made so that it can be applied to a series of facts from physics and astronomy. Further, the point of view acquired in the specific example of the top will be applied to the conception of mechanics in general, and finally some detailed excursions into the domain of modern theoretical physics will be undertaken. xi

12

13 Volume II Development of the Theory in the Case of the Heavy Symmetric Top

14