The Theory of the Top. Volume II (eBook)

Contents 6
Advertisement of Volume II of the Theory of the Top 10
Volume II Development of the Theory in the Case of the Heavy Symmetric Top 14
Chapter IV. The general motion of the heavy symmetric top. Introduction to elliptic integrals. 16
§1. Intuitive discussion of the expected forms of motion preliminary agreements.
§2. Intuitive discussion of the expected forms of motion continuation and conclusion.
§3. Quantitative treatment of the general motion of the heavy symmetric top. Execution of the six required integrations. 35
§4. General periodicity properties of the motion. Preliminaries on the behavior of the elliptic integrals for a circulation of a, ß, ., d. 43
§5. On the relation between the motions of different tops that yield the same impulse curve, and on the motion of the spherical top. 50
§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.
§7. The characteristic curves of the third order for arbitrary position of the initial circle e distinction between strong and weak tops.
§8. On the numerical calculation of the elliptic integrals for t and .. 78
§9. On the approximate calculation of the top trajectories. 88
Chapter V. On special forms of motion of the heavy symmetric top, particularly pseudoregular precession, and on the stability of motion. 98
§1. Regular precession and its neighboring forms of motion. 98
§2. Pseudoregular precession resolution of the paradoxes of the motion of the top.
§3. Popular explanations of the phenomena of the top in the literature. 126
§4. On the stability of the upright top. Geometric discussion. 135
§5. Continuation. Analytic treatment of the motion of the upright top altered by an impact.—Formulas for pseudoregular precession with small precession circle. 145
§6. Generalities on the stability and lability of motion. 161
§7. Energy criteria for the stability of equilibrium andmotion. 173
§8. On the method of small oscillations. 183
§9. On the motion of the heavy asymmetric top. 193
Chapter VI. Representation of the motion of the top by elliptic functions. 211
§1. The Riemann surface (u,vU). 211
§2. Behavior of the elliptic integrals on the Riemann surface. 216
§3. The image of the Riemann surface (u,vU) in the t-plane. 225
§4. Representation of a, ß, ., d by v-quotients. 236
§5. The trajectory of the apex of the top, the polhode and herpolhode curves, etc., represented by v-quotients. 249
§6. Numerical calculation of the motion by v-series. 259
§7. Representation of the motion of the force-free top by elliptic functions. 273
§8. Conjugate Poinsot motions. Jacobi’s theorem on the relation between the motion of the force-free asymmetric top and the heavy spherical top. 295
§9. The Lagrange equations for a, ß, ., d of the heavy spherical top and their direct integration. Relation between the motion of the spherical top and a problem in particle mechanics. 310
Appendix to Chapter VI. 332
§10. The top on the horizontal plane. 332
Addenda and Supplements. To Chap. V. 351
Translators’ Notes. 361
References. 411
Index 421