Constants of the motion for nonslipping tippe tops and other tops with round pegs

American Journal of Physics

Volumn 68, Issue 9, 2000, Pages 821-828

  Gray, C G ^a   Nickel, B G ^a  

^a UNIVERSITY OF GUELPH   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034404237     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1302299     Document Type: Article

References (64)

1. The history of the theory of motion of solid bodies on a plane surface is sketched by Routh (Ref. 40, Pt. 2, p. 186) for the cases of no friction, and perfect friction (i.e., no slipping). For the case of sliding friction, according to Perry (Ref. 2, p. 39) and Gray (Ref. 31, p. 393), the earliest work is due to A. Smith and Kelvin in the 1840s. The work of Gallop (Ref. 3) and Jellett (Ref. 34) is also seminal.
2. J. Perry, Spinning Tops and Gyroscopic Motions (Sheldon, London, 1890, reprinted by Dover, 1957), p. 43.
3. E. G. Gallop, "On the Rise of a Spinning Top," Trans. Cambridge Philos. Soc. 19, 356-373 (1904). Some of Gallop's results are summarized in Deimel (Ref. 32, p. 93) and Gray (Ref. 31, p. 396).
4. H. Crabtree, An Elementary Treatment of the Theory of Spinning Tops and Gyroscopic Motion (Longmans Green, London, 1909, reprinted by Chelsea, 1967), p. 5.
5. A. D. Fokker, "The Tracks of Tops Pegs on the Floor," Physica (Amsterdam) 18, 497-502 (1952), see also "The Rising Top, Experimental Evidence and Theory," 8, 591-596 (1941).
6. A. D. Fokker, "The Tracks of Tops Pegs on the Floor," Physica (Amsterdam) 18, 497-502 (1952), see also "The Rising Top, Experimental Evidence and Theory," 8, 591-596 (1941).
7. C. M. Braams, "On the Influence of Friction on the Motion of a Top," Physica (Amsterdam) 18, 503-514 (1952); "The Symmetrical Spherical Top," Nature (London) 170, 31 (1952); "The Tippe Top," Am. J. Phys. 22, 568 (1954).
8. C. M. Braams, "On the Influence of Friction on the Motion of a Top," Physica (Amsterdam) 18, 503-514 (1952); "The Symmetrical Spherical Top," Nature (London) 170, 31 (1952); "The Tippe Top," Am. J. Phys. 22, 568 (1954).
9. C. M. Braams, "On the Influence of Friction on the Motion of a Top," Physica (Amsterdam) 18, 503-514 (1952); "The Symmetrical Spherical Top," Nature (London) 170, 31 (1952); "The Tippe Top," Am. J. Phys. 22, 568 (1954).
10. N. M. Hugenholtz, "On Tops Rising by Friction," Physica (Amsterdam) 18, 515-527 (1952).
11. J. A. Haringx, "De Wondertol," De Ingenieur 4, 13-17 (1952).
12. J. A. Jacobs, "Note on the Behaviour of a Certain Symmetrical Top," Am. J. Phys. 20, 517-518 (1952).
13. D. Van Ostenburg and C. Kikuchi, "Some Analogies of the Tippe Top to Electrons and Nuclei," Am. J. Phys. 21, 574 (1953).
14. F. Schuh, "Beweging van een Excentrisch Bezaarde Bol Over een Horizontaal Vlak in Verband met de Tovertol 'Tippe Top,'" Proc. K. Ned. Akad. Wet., Ser. A: Math. Sci. 56, 423-452 (1953).
15. W. A. Pliskin, "The Tippe Top (Topsy-Turvy Top)," Am. J. Phys. 22, 28-32 (1954).
16. S. O'Brien and J. L. Synge, "The Instability of the Tippe Top Explained by Sliding Friction," Proc. R. Ir. Acad. Sect. A, Math. Astron. Phys. Sci. 56, 23-35 (1954).
17. A. R. Del Campo, "Tippe Top (Topsy-Turvee Top) Continued," Am. J. Phys. 23, 544-545 (1955).
18. I. M. Freeman, "The Tippe Top Again," Am. J. Phys. 24, 178 (1956).
19. D. G. Parkyn, "The Inverting Top," Math. Gazette 40, 260-265 (1956).
20. W. Schallreuter, "Der Spielkreisel mit Selbstaufrichtung." Wiss. Z. der Ernst Moritz Arndt-Universität Greifswald 8, 43-46 (1958-9).
21. D. G. Parkyn, "The Rising of Tops with Rounded Pegs," Physica (Amsterdam) 24, 313-330 (1958).
22. L. S. Isaeva, "On the Sufficient Conditions of Stability of Rotation of a Tippe-Top on a Perfectly Rough Horizontal Surface," J. Appl. Math. Mech. 23, 403-406 (1959).
23. J. B. Hart, "Angular Momentum and Tippe Top," Am. J. Phys. 27, 189 (1959).
24. F. Johnson, "The Tippy Top," Am. J. Phys. 28, 406-407 (1960).
25. G. D. Freier, "The Tippy-Top," Phys. Teach. 5, 36-38 (1967).
26. For a delightful picture of Bohr and Pauli playing with a tippe top, see S. Rozental, ed., Niels Bohr: His Life and Work as Seen by His Friends and Colleagues (Interscience, New York, 1967), p. 209.
27. J. C. Lauffenburger, "A Large-Scale Demonstration of the Tippe Top," Am. J. Phys. 40, 1338 (1972).
28. R. J. Cohen, "The Tippe Top Revisited," Am. J. Phys. 45, 12-17 (1977).
29. J. Walker, The Flying Circus of Physics (Wiley, New York, 1977), p. 43.
30. T. R. Kane and D. A. Levinson, "A Realistic Solution of the Symmetric Top Problem," J. Appl. Mech. 45, 903-909 (1978).
31. N. G. Chataev, Theoretical Mechanics (MIR, Moscow, 1987; revised edition by Springer, Berlin, 1989), p. 178. Chataev discusses the very similar Chinese top.
32. V. D. Barger and M. G. Olsson, Classical Mechanics: A Modern Perspective (McGraw-Hill, New York, 1995), 2nd ed., p. 273.
33. H. Leutwyler, "Why Some Tops Tip," Eur. J. Phys. 15, 59-61 (1994).
34. A. Gray, A Treatise on Gyrostatics and Rotational Motion (MacMillan, London, 1918, reprinted by Dover, 1959).
35. R. H. Deimel, Mechanics of the Gyroscope (MacMillan, New York, 1929; reprinted by Dover, 1950).
36. z is reduced (conservation of energy - the center of mass has risen), so that a z torque, which can only arise from friction, must have played a role. To see that some sliding friction is essential, note (ii) conservative systems in bounded motions generally oscillate between turning points, and do not in general reach a final steady state as observed in Fig. 1(b), and (iii) a pure rolling motion (with pivoting allowed but no sliding) would produce in Fig. 1(b) the top with angular momentum pointing down, rather than still up as in Fig. 1(b). This can be seen by visualizing the top motion, or by slowly rolling an actual top. [The modern way of stating (ii) is "Conservative Systems Have No Attractors," see, e.g., R. L. Borrelli and C. S. Coleman, Differential Equations (Wiley, New York, 1998), p. 457.]
37. J. H. Jellett, A Treatise on the Theory of Friction (MacMillan, London, 1872), p. 185.
38. We consider only sliding kinetic friction here, the most important type for the tippe top. For tops in general, there is also rolling friction (due to finite elasticity of the surfaces in contact), boring (or pivoting) friction (due to finite sized area of contact), and air friction.
39. M. Tabor, Chaos and Integrability in Nonlinear Dynamics (Wiley, New York, 1989).
40. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1989), 2nd ed.
41. G = R - a cos θ), and the three Euler angles giving the orientation. There are two independent nonholonomic (rolling), or local, constraints, thus giving three local degrees of freedom.
42. We have found no references to Routh's discussion of the integrability of the nonsliding tippe top. Even a recent monograph devoted to integrable top systems does not mention it: M. Audin, Spinning Tops: A Course on Integrable Systems (Cambridge U. P., New York, 1996).
43. E. J. Routh, A Treatise on the Dynamics of a System of Rigid Bodies, Pt. 1. The Elementary Part (MacMillan, London, 1860, 1905), 6th ed., Pt. 2, The Advanced Part (reprinted by Dover, New York, in 1960 and 1955, respectively). See especially Pt. 2. p. 192.
44. E. J. Routh, A Treatise on the Dynamics of a System of Rigid Bodies, Pt. 1. The Elementary Part (MacMillan, London, 1860, 1905), 6th ed., Pt. 2, The Advanced Part (reprinted by Dover, New York, in 1960 and 1955, respectively). See especially Pt. 2. p. 192.
45. Edward John Routh (1831-1907) is remembered for the Routhian (Ref. 42), the Routh-Hurwitz and other stability criteria (Ref. 43), and the Routh rules (Ref. 44) for moments of inertia, as well as for his books (Ref. 40). Interestingly, in the Cambridge Mathematical Tripos examination of 1854 Routh finished first (senior wrangler), and Maxwell second. His books have been out of fashion for some time (Ref. 45), but are an invaluable source of results and inspiration.
46. H. Goldstein, Classical Mechanics (Addison-Wesley, Cambridge, MA. 1980), 2nd ed., p. 352.
47. D. R. Merkin, Introduction to the Theory of Stability (Springer, New York, 1997), pp. 84, 111.
48. J. L. Synge and B. A. Griffith, Principles of Mechanics (McGraw-Hill, New York, 1959), 2nd ed., p. 293.
49. According to William Fogg Osgood, author of a number of textbooks, including Mechanics (MacMillan, London, 1937; reprinted by Dover, 1965), "Routh's exposition of the theory is execrable, but his lists of problems, garnered from the old Cambridge Tripos papers, are capital." (Osgood, p. 246).
50. R. Baierlein, Newtonian Dynamics (McGraw-Hill, New York, 1983), Chap. 7.
51. E. A. Milne, Vectorial Mechanics (Methuen, London, 1948), Chaps. 15-17.
52. W. Case, "The Gyroscope: An Elementary Discussion of a Child's Toy," Am. J. Phys. 45, 1107-1109 (1977); W. B. Case and M. A. Shay, "On the Interesting Behaviour of a Gimbal-mounted Gyroscope," ibid. 60, 503-506 (1992).
53. W. Case, "The Gyroscope: An Elementary Discussion of a Child's Toy," Am. J. Phys. 45, 1107-1109 (1977); W. B. Case and M. A. Shay, "On the Interesting Behaviour of a Gimbal-mounted Gyroscope," ibid. 60, 503-506 (1992).
54. The three Euler angles are θΦψ, with Φ describing the azimuth position around Z, and ψ the spin position around 3. Since there are X, Y and 1, 2 rotational symmetries in the problem, Φ and ψ will contain arbitrary reference values.
55. For a masterful account of nonholonomic constraints, see J. G. Papastavridis, Tensor Calculus and Analytical Dynamics (CRC Press, Boca Raton, FL, 1999).
56. O=ωXRẑ, since point C is always directly under point O (see Fig. 2), and v=ωXr, we again get (11).
57. C the velocity of the contact point as traced out on the horizontal surface (see Ref. 51).
58. See, for example, Osgood (Ref. 45, p. 456), Synge and Griffith (Ref. 44, pp. 328, 388) Goldstein (Ref. 42, pp. 75, 76, 215, 216, or Chataev (Ref. 28, pp. 130, 136, 210).
59. D. J. McGill and J. G. Papastavridis, "Comments on 'Comments on Fixed Points in Torque-Angular Momentum Relations,'" Am. J. Phys. 55, 470-471 (1987).
60. z(θ) =Mg+Ma(1/2 sin θf′(θ)+cos θf(θ)).
61. y given in the text.
62. z would project a large component along the 3 axis. In other words, the center of mass has risen.
63. r̂. Whether this has any significance for the rolling case is not clear.
64. Routh (Ref. 40), p. 194.