The Theory of the Top. Volume II: Development of the Theory in the Case of the Heavy Symmetric Top / Edition 1

The Theory of the Top. Volume II: Development of the Theory in the Case of the Heavy Symmetric Top / Edition 1

ISBN-10:
0817648240
ISBN-13:
9780817648244
Pub. Date:
07/22/2010
Publisher:
Birkhäuser Boston

Hardcover

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Overview

The Theory of the Top. Volume II: Development of the Theory in the Case of the Heavy Symmetric Top / Edition 1

The Theory of the Top. Volume II. Development of the Theory in the Case of the Heavy Symmetric Top is the second in a series of four self-contained English translations of the classic and definitive treatment of rigid body motion.

Key features:

* Complete and unabridged presentation with recent advances and

additional notes

* Annotations by the translators provide insights into the nature of

science and mathematics in the late 19th century

* Each volume interweaves theory and applications

The first volume established the general kinematic and kinetic foundations of the theory. Volume II discusses the motion of the symmetric top with a fixed support point, under the influence of gravity, in all its details. The Theory of the Top was originally presented by Felix Klein as an 1895 lecture at Göttingen University that was broadened in scope and clarified as a result of collaboration with Arnold Sommerfeld.

Graduate students and researchers interested in theoretical and applied mechanics will find this a thorough and insightful account.

Other works in this series include Volume I. Introduction to the Kinematics and Kinetics of the Top, Volume III. Perturbations. Astronomical and Geophysical Applications, and Volume IV. Technical Applications of the Theory of the Top.

Product Details

ISBN-13: 9780817648244
Publisher: Birkhäuser Boston
Publication date: 07/22/2010
Edition description: 2010
Pages: 414
Product dimensions: 6.40(w) x 9.30(h) x 1.20(d)

Table of 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; preliminary agreements 197

§2 Intuitive discussion of the expected forms of motion; continuation and conclusion 204

§3 Quantitative treatment of the general motion of the heavy symmetric top. Execution of the six required integrations 216

§4 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 224

§5 On the relation between the motions of different tops that yield the same impulse curve, and on the motion of the spherical top. 231

§6 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 = 0. 239

§7 The characteristic curves of the third order for arbitrary position of the initial circle e; distinction between strong and weak tops 247

§8 On the numerical calculation of the elliptic integrals for t and ψ 259

§9 On the approximate calculation of the top trajectories 269

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 279

§2 Pseudoregular precession; resolution of the paradoxes of the motion of the top 291

§3 Popular explanations of the phenomena of the top in the literature 307

§4 On the stability of the upright top. Geometric discussion 316

§5 Continuation. Analytic treatment of the motion of the upright top altered by an impact.-Formulas for pseudoregular precession with small precession circle 326

§6 Generalities on the stability and lability of motion 342

§7 Energy criteria for the stability of equilibrium and motion 354

§8 On the method of small oscillations 364

§9 On the motion of the heavy asymmetric top. 374

Chapter VI Representation of the motion of the top by elliptic functions.

§1 The Riemann surface (u, √U) 392

§2 Behavior of the elliptic integrals on the Riemann surface 397

§3 The image of the Riemann surface (u, √U) in the t-plane 406

§4 Representation of α, β, γ, δ by v-quotients 417

§5 The trajectory of the apex of the top, the polhode and herpolhode curves, etc., represented by v-quotients 430

§6 Numerical calculation of the motion by v-series 440

§7 Representation of the motion of the force-free top by elliptic functions 454

§8 Conjugate Poinsot motions. Jacobi's theorem on the relation between the motion of the force-free asymmetric top and the heavy spherical top 476

§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 491

Appendix to Chapter VI.

§10 The top on the horizontal plane 513

Addenda and Supplements 533

Translators' Notes 543

References 593

Index 603

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