New and more illuminating derivations are given for the three constants of the motion of a nonsliding tippe top and other symmetric tops with a spherical peg in contact with a horizontal plane. Some rigorous conclusions about the motion can be drawn immediately from these constants. It is shown that the system is integrable, and provides a valuable pedagogical example of such systems. The equation for the tipping rate is reduced to one-dimensional form. The question of sliding versus nonsliding is considered. A careful literature study of the work over the past century on this problem has been done. The classic work of Routh has been rescued from obscurity, and some misstatements in the literature are corrected.
REFERENCES
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
).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
),A. D.
Fokker
, see also “The Rising Top, Experimental Evidence and Theory
,” Physica (Amsterdam)
8
, 591
–596
(1941
).6.
C. M.
Braams
, “On the Influence of Friction on the Motion of a Top
,” Physica (Amsterdam)
18
, 503
–514
(1952
);C. M.
Braams
, “The Symmetrical Spherical Top
,” Nature (London)
170
, 31
(1952
);7.
N. M.
Hugenholtz
, “On Tops Rising by Friction
,” Physica (Amsterdam)
18
, 515
–527
(1952
).8.
9.
J. A.
Jacobs
, “Note on the Behaviour of a Certain Symmetrical Top
,” Am. J. Phys.
20
, 517
–518
(1952
).10.
D.
Van Ostenburg
and C.
Kikuchi
, “Some Analogies of the Tippe Top to Electrons and Nuclei
,” Am. J. Phys.
21
, 574
(1953
).11.
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
).12.
W. A.
Pliskin
, “The Tippe Top (Topsy-Turvy Top)
,” Am. J. Phys.
22
, 28
–32
(1954
).13.
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
).14.
A. R.
Del Campo
, “Tippe Top (Topsy-Turvee Top) Continued
,” Am. J. Phys.
23
, 544
–545
(1955
).15.
16.
D. G.
Parkyn
, “The Inverting Top
,” Math. Gazette
40
, 260
–265
(1956
).17.
W.
Schallreuter
, “Der Spielkreisel mit Selbstaufrichtung
,” Wiss. Z. der Ernst Moritz Arndt-Universität Greifswald
8
, 43
–46
(1958
–9).18.
D. G.
Parkyn
, “The Rising of Tops with Rounded Pegs
,” Physica (Amsterdam)
24
, 313
–330
(1958
).19.
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
).20.
J. B.
Hart
, “Angular Momentum and Tippe Top
,” Am. J. Phys.
27
, 189
(1959
).21.
22.
23.
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.
24.
J. C.
Lauffenburger
, “A Large-Scale Demonstration of the Tippe Top
,” Am. J. Phys.
40
, 1338
(1972
).25.
R. J.
Cohen
, “The Tippe Top Revisited
,” Am. J. Phys.
45
, 12
–17
(1977
).26.
J. Walker, The Flying Circus of Physics (Wiley, New York, 1977), p. 43.
27.
T. R.
Kane
and D. A.
Levinson
, “A Realistic Solution of the Symmetric Top Problem
,” J. Appl. Mech.
45
, 903
–909
(1978
).28.
N. G. Chataev, Theoretical Mechanics (MIR, Moscow, 1987; revised edition by Springer, Berlin, 1989), p. 178. Chataev discusses the very similar Chinese top.
29.
V. D. Barger and M. G. Olsson, Classical Mechanics: A Modern Perspective (McGraw–Hill, New York, 1995), 2nd ed., p. 273.
30.
31.
A. Gray, A Treatise on Gyrostatics and Rotational Motion (MacMillan, London, 1918, reprinted by Dover, 1959).
32.
R. H. Deimel, Mechanics of the Gyroscope (MacMillan, New York, 1929; reprinted by Dover, 1950).
33.
Some qualitative arguments may be helpful here. (i) Friction of some type is necessary for the top to make the transition shown in Fig. 1, since 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.]
34.
J. H. Jellett, A Treatise on the Theory of Friction (MacMillan, London, 1872), p. 185.
35.
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.
36.
M. Tabor, Chaos and Integrability in Nonlinear Dynamics (Wiley, New York, 1989).
37.
V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1989), 2nd ed.
38.
We say local because the tippe top has five global degrees of freedom corresponding to the X,Y coordinates of the center of mass G (the Z coordinate is constrained by ), and the three Euler angles giving the orientation. There are two independent nonholonomic (rolling), or local, constraints, thus giving three local degrees of freedom.
39.
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).
40.
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.
41.
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.
42.
H. Goldstein, Classical Mechanics (Addison–Wesley, Cambridge, MA, 1980), 2nd ed., p. 352.
43.
D. R. Merkin, Introduction to the Theory of Stability (Springer, New York, 1997), pp. 84, 111.
44.
J. L. Synge and B. A. Griffith, Principles of Mechanics (McGraw–Hill, New York, 1959), 2nd ed., p. 293.
45.
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).
46.
R. Baierlein, Newtonian Dynamics (McGraw–Hill, New York, 1983), Chap. 7.
47.
E. A. Milne, Vectorial Mechanics (Methuen, London, 1948), Chaps. 15–17.
48.
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
,” Am. J. Phys.
60
, 503
–506
(1992
).49.
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.
50.
For a masterful account of nonholonomic constraints, see J. G. Papastavridis, Tensor Calculus and Analytical Dynamics (CRC Press, Boca Raton, FL, 1999).
51.
Note that The correct relation is where is the vector from the origin of an arbitrary set of space fixed axes to the moving point C. Using since point C is always directly under point O (see Fig. 2), and we again get (11).
52.
Isaeva (Ref. 19) states “The projection of the angular momentum about the point of contact on a principal axis of inertia is a constant.” We are unable to confirm this claim. The angular momentum about contact point C, is related to that about G, by Because we have as another version of the Jellet constant (10). This gives where Hence and are related. However, the individual quantities and can be calculated from [using an equation of motion (Ref. 54) which differs from (2), i.e., ], and do not vanish. [An exception occurs in the limit —see the discussion below Eq. (29).] Here is the torque about C, and the velocity of the contact point as traced out on the horizontal surface (see Ref. 51).
54.
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
).55.
The normal force is given by [see Fig. 2 or Eq. (1)] Since and we have From we obtain so that as a function of θ is
56.
We choose X,Y axes as in Fig. 4. can be obtained from [see the argument above Eq. (13)] From (31) we get Evaluating from (32) then yields expression (38) for given in the text. To obtain we start with so that Using (see Fig. 4) and we get We then use to get expression (39) for given in the text.
57.
The full argument (Ref. 34) is somewhat lengthy, but the essence is as follows. Since [see (13)] any change in the initial is only a small fraction (∼7% for our top) of the change in Suppose the top starts as in Fig. 1(a), with and both large. If after some time the sliding friction has reduced to a small value, say zero, is thus reduced to about 93% of its initial value. This means θ must have increased to about 90°, otherwise would project a large component along the 3 axis. In other words, the center of mass has risen.
58.
A hint is perhaps contained in a result of Gallop (Ref. 3) [see also Deimel (Ref. 32) and Leutwyler (Ref. 30)]. For the sliding case, where no longer holds (see Appendix B), Gallop shows that the minimum of the total kinetic energy, at fixed θ and fixed Jellett constant J (13), is where with is the effective moment of inertia occurring in Routh’s constant (25) when rewritten as The minimum occurs with the top rotating about and the corresponding moment of inertia is related to by Whether this has any significance for the rolling case is not clear.
59.
Routh (Ref. 40), p. 194.
This content is only available via PDF.
© 2000 American Association of Physics Teachers.
2000
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.