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.

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
),
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
);
C. M.
Braams
, “
The Tippe Top
,”
Am. J. Phys.
22
,
568
(
1954
).
7.
N. M.
Hugenholtz
, “
On Tops Rising by Friction
,”
Physica (Amsterdam)
18
,
515
527
(
1952
).
8.
J. A.
Haringx
, “
De Wondertol
,”
De Ingenieur
4
,
13
17
(
1952
).
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.
I. M.
Freeman
, “
The Tippe Top Again
,”
Am. J. Phys.
24
,
178
(
1956
).
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.
F.
Johnson
, “
The Tippy Top
,”
Am. J. Phys.
28
,
406
407
(
1960
).
22.
G. D.
Freier
, “
The Tippy-Top
,”
Phys. Teach.
5
,
36
38
(
1967
).
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.
H.
Leutwyler
, “
Why Some Tops Tip
,”
Eur. J. Phys.
15
,
59
61
(
1994
).
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 ωz 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 (xG,yG,θ,φ,ψ) corresponding to the X,Y coordinates of the center of mass G (the Z coordinate is constrained by zG=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.
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 v. The correct relation is =vC, where rC is the vector from the origin of an arbitrary set of space fixed axes to the moving point C. Using C=vO=ω×Rẑ, since point C is always directly under point O (see Fig. 2), and v=ω×r, 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, LC, is related to that about G, LGL, by LC=L+Mr×v. Because LCr=Lr, we have LCr=const as another version of the Jellet constant (10). This gives LCZ−(a/R)LC3=const, where LCZ=LC3cos θ+LC2sinθ. Hence LC3 and LC2 are related. However, the individual quantities C1,C2, and C3 can be calculated from C [using an equation of motion (Ref. 54) which differs from (2), i.e., C=τC+Mv×C], and do not vanish. [An exception occurs in the limit R→0—see the discussion below Eq. (29).] Here τC is the torque about C, and C the velocity of the contact point as traced out on the horizontal surface (see Ref. 51).
53.
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).
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)] Fz=Mg+Mv̇z. Since z=z̈ and z=R−a cos θ, we have z=a sin θθ̈+a cos θθ̇2. From θ̇2=f(θ) we obtain θ̈=f(θ)/2, so that Fz as a function of θ is Fz(θ)=Mg+Ma(12sin θf(θ)+cos θf(θ)).
56.
We choose X,Y axes as in Fig. 4. Fx≡F1 can be obtained from [see the argument above Eq. (13)] FxR sin θ=τ3=I3ω̇3. From (31) we get ω̇3=−KR3/f32. Evaluating 3 from (32) then yields expression (38) for Fx given in the text. To obtain Fy we start with τ=, so that τ⋅1̂=⋅1̂=(d/dt)(L⋅1̂)−L⋅1̂̇=I1ω̇1L⋅1̂̇. Using (see Fig. 4) 1̂̇=φ̇ŷ=(ω2/sin θ)ŷ=(ω2/sin θ)(cos θ2̂sin θ3̂), and τ⋅1̂=Fy(R−a cos θ)−Fza sin θ, we get Fy(R−a cos θ)−Fza sin θ=I1ω̇1−I1cot θω22+I3ω2ω3. We then use ω̇1≡θ̈=f(θ)/2 to get expression (39) for Fy given in the text.
57.
The full argument (Ref. 34) is somewhat lengthy, but the essence is as follows. Since [see (13)] z=(a/R)L̇3, any change in the initial Lz is only a small fraction (a/R) (∼7% for our top) of the change in L3. Suppose the top starts as in Fig. 1(a), with θ≈0 and Lz≈L3 both large. If after some time the sliding friction has reduced L3 to a small value, say zero, Lz is thus reduced to about 93% of its initial value. This means θ must have increased to about 90°, otherwise Lz 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 v=ω×r 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 Kmin(J)(θ)=J2/2Ieff(θ), where Ieff(θ)=I1sin2 θ+I3(cos θ−ε)2, with ε=a/R, is the effective moment of inertia occurring in Routh’s constant (25) when rewritten as [(1/2)(I3I1/MR2)+(1/2)Ieff(θ)]ω32=const. The minimum occurs with the top rotating about r, and the corresponding moment of inertia I is related to Ieff(θ) by Ieff(θ)=(r/R)2 I. 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.
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.