Constants of the motion for nonslipping tippe tops and other tops with round pegs C. G. Graya) and B. G. Nickelb) Department of Physics, University of Guelph, Guelph, Ontario NIG 2W1, Canada ͑Received 5 August 1999; accepted 21 December 1999͒ 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. © 2000 American Association of Physics Teachers.
I. INTRODUCTION shows that there are three constants of the motion: the en- ergy, the so-called Jellett constant ͑a linear combination of The behavior of the tippe top has fascinated physicists for angular momentum components͒, and what we term the over a century.1–30 Textbook discussions are, however, still Routh constant. The physical significance of the Routh con- quite rare and brief.26,28,29 Figure 1 shows schematically the stant has not yet been understood, but we show that it in- small wooden commercial model readily available. ͑Appen- volves purely kinetic energy. dix A gives the specifications of the model in our posses- Our contribution is to simplify the derivation of the con- sion.͒ This type was introduced in Denmark a half century stants of the motion, and to discuss their physical signifi- ago. In the nineteenth century loaded ͑eccentric͒ spheres of cance as currently understood. The simplifications come various kinds were used2–4 ͑Fig. 2͒. Here we discuss some from using vector methods and from working with a simpler aspects of the motion of such tippe tops, but the results are in set of angular velocity components than the Euler angle de- fact more general, and apply to any symmetric top with a rivatives. We use the standard angular velocity components spherical peg in contact with a horizontal plane. referred to the body-fixed principal axes. There are certainly It has been established,33 both experimentally5,12,18,21 and problems in rigid body motion where the introduction of the ͑ theoretically,3,6,7,27,34 that a full explanation of the steady ris- Euler angles is advantageous e.g., asymmetric tops, steady ͒ 46–48 ing of the center of gravity of both ordinary and tippe tops precessional motion, etc. , but there are cases, particu- ͑ ͒ requires sliding friction.35 There are regimes, however, larly involving symmetric tops such as the tippe top , where where sliding can be neglected and the top rolls and pivots, the calculations are simpler without them. We also have e.g., when sufficient energy has been dissipated by sliding, or carefully studied the literature of this problem for the past if the top is started with low kinetic energy. Experiments5,18 century, point out some overlooked references, and correct and computer simulations27 have shown that, for both ordi- some misstatements. The nonslipping tippe top can serve a nary and tippe tops, some motions consist of alternating roll- useful pedagogical role in advanced mechanics courses, in ing and sliding segments. The nondissipative limiting cases either the rigid body motion section or the section dealing of pure rolling, and sliding without friction ͑Appendix B͒, with integrability and chaos. can serve as zeroth-order approximations for perturbation The paper is organized as follows: The translational and treatments,6,7,12,16 with the sliding friction as the perturba- rotational equations of motion are given in Sec. II, followed tion. It is the rolling regimes we study here—for two rea- by three sections giving the derivations of the three constants sons. First, the results have relevance to the real top motion of the motion, and then a discussion. There are also two as just explained, and second, in the nonsliding regime the appendices, one on the frictionless case, and another giving tippe top provides a beautiful nontrivial example of an inte- some data for the tippe top in our possession. grable system. The study of such systems is much in vogue,36,37 since they are rare. As is well known, most non- linear systems ͑as is the tippe top͒ are nonintegrable, and can II. EQUATIONS OF MOTION display chaotic motion. An integrable system has at least as many independent constants of the motion as there are de- For our purposes the older model tippe top is slightly more grees of freedom, and often ͑as here͒ the dynamics is sepa- convenient, i.e., a loaded ͑eccentric͒ sphere. The center of rable, i.e., can be reduced to a set of one-dimensional prob- mass is off center by a distance a—see Fig. 2. There are no lems. Integrable systems can display only regular ͑i.e., qualitative differences in the behaviors of the tops of Figs. 1 7,16 nonchaotic͒ motions. As we shall see, with perfect friction and 2, and the minor differences are easily explained. ͑i.e., no slipping͒, there are three ͑local38͒ degrees of free- Figure 2 shows the geometry. As is standard in rigid body dom and three constants of the motion. ͓Another limiting theory, we use two sets of axes: space-fixed axes ͑XYZ͒ with ͑ ͒ case—no friction—is also integrable, and is much easier to Z vertical, and body-fixed axes 123 with 3 along the top ͑ ͒ discuss ͑Appendix B͒.͔ symmetry axis and origin at the center of mass G . The only 49 A classic, but totally ignored,39 discussion of the integra- angle of interest to us is , the angle between the Z and 3 bility of the nonsliding tippe top was given by Routh.40,41 He axes. Its rate of change, ˙ , gives the tipping rate.
821 Am. J. Phys. 68 ͑9͒, September 2000 http://ojps.aip.org/ajp/ © 2000 American Association of Physics Teachers 821 Fig. 1. Schematic of the tippe top ͑construction details are in Appendix A͒. Given a spin on its spherical side, it quickly turns over and spins on the stem, the center of gravity thus rising ͑from Ref. 26͒.
The translational and rotational equations of motion for the top are Mv˙ϭFtotalϵFϩw, ͑1͒ and Fig. 3. Special choice of 1,2 axes, denoted 1Ј,2Ј. Here 2ˆЈ is in the plane of the page ͑the ZO3 plane͒, and 1ˆЈ ͑along the so-called line of nodes͒ is L˙ ϭ ϵϪrϫF, ͑2͒ perpendicular to the page and outward, such that 1Ј 2Ј 3 is a right-handed system. respectively, where M is the mass, v the center of mass ve- ϵ total locity, • d/dt is the time rate of change, F is the total external force on the top, F the reaction force at contact point is nonholonomic50 ͑i.e., a nonintegrable, or rolling con- C, and wϭϪMgzˆ the vertically acting weight. L is the an- straint͒, and is the condition for no slip at contact point C; gular momentum with respect to the center of mass, and Eq. ͑4͒ states that the top instantaneously rotates about point ϭϪrϫF is the torque about the center of mass due to F ͑the C. Here is the angular velocity of the top. Equations ͑3͒ minus sign arising because our r, the relative position of G and ͑4͒ allow three local degrees of freedom.38 ϭ with respect to C, points from C to G͒. For a symmetric top (I2 I1) the angular velocity and an- There are also two constraint conditions. The first is holo- gular momentum are related by nomic ͑integrable, or geometric͒, i.e., ϭ ˆ ϩ ˆ ϩ ˆ ϭ ϩ⌬ ˆ ͑ ͒ L I1 11 I1 22 I3 33 I1 I 33, 5 rϭRzˆϪa3ˆ , ͑3͒ ˆ where we have added and subtracted a term I1 33, and which is the condition for the top to always be in contact where the are body-fixed principal axes components with with the surface. The second, i 1ˆ ,2ˆ , and 3ˆ , unit vectors along the principal axes. The prin- vϭϫr, ͑4͒ ⌬ ϭ Ϫ cipal moments of inertia are (I1 ,I1 ,I3) and I I3 I1 . We note there are two dimensionless asymmetry param- eters in the problem: the eccentricity a/R, and the anisotropy ⌬ I/I1 . For tippe tops a/R will be positive and small, whereas for ordinary tops it will be negative and usually large. Appendix A gives the order of magnitude of the asym- metry parameters for our particular top, a tippe top. We now discuss, in turn, the three integrals, or constants of the motion, for the equations of motion ͑1͒ and ͑2͒.
III. THE ENERGY CONSTANT Because we assume rolling without slipping, there is no energy dissipation, so that an obvious constant of the motion ϭ ϩ ϵ ϩ ϩ is the total energy E K V Krot Ktr V, where Krot ϭ 1 2ϩ 1 2ϩ 1 2 2 I1 1 2 I1 2 2 I3 3 is the rotational kinetic energy, ϭ 1 2 Ktr 2 Mv is the translational kinetic energy, and V ϭMga(1Ϫcos ) is the gravitational potential energy. Our convention for V is that Vϭ0 when ϭ0, i.e., when the ϭ Ϫ Fig. 2. Loaded sphere version of the tippe top. The center of mass G is off center of mass Z coordinate is z (R a). To simplify the ͑ ͒ ͑ ϭ 1 ϫ 2ϭ 1 ͓ ϫ 2ϩ ϫ 2 center O by distance a. Space-fixed axes are XYZ with Z vertical X and Y expression for Ktr 2 M( r) 2 M ( r)1 ( r)2 ͒ can be chosen arbitrarily and are not shown , and body-fixed axes are 123 ϩ(ϫr)2͔, we choose 1, 2 axes as in Fig. 3, and denote this with 3 along the symmetry axis ͑1 and 2 can be chosen arbitrarily and are 3 special choice as 1Ј,2Ј. The 1Ј,2Ј axes are not body-fixed not shown͒; zˆ and 3ˆ are unit vectors along Z and 3, respectively. The forces acting are the weight wϭϪMgzˆ at G, and the reaction force F at contact axes, but they are principal axes, so that the expression for point C. F has a normal component F , and two transverse frictional com- Krot given above is still valid. In this coordinate system, we z ϭ ϭ ϭ Ϫ ponents Fʈ and FЌ , which are parallel and perpendicular, respectively, to have r1Ј 0, r2Ј R sin , and r3 R cos a. This gives the plane of the paper ͑i.e., the ZO3 plane͒. The friction forces are static for ϭ 1 22 ϩ 1 ͑͑ Ϫ ͒ Ϫ ͒2 pure rolling. rϭRzˆϪa3ˆ is the vector from C to G. Ktr 2 Mr 1Ј 2 M R cos a 2Ј R sin 3 ,
822 Am. J. Phys., Vol. 68, No. 9, September 2000 C. G. Gray and B. G. Nickel 822 2ϭ 2 ϩ 2ϭ 2ϩ 2Ϫ where r r2Ј r3 R a 2Ra cos . The expression for the energy is thus 1 ͓ ϩ ͑ 2ϩ 2Ϫ ͔͒2 ϩ 1 2 2 I1 M R a 2Ra cos 1Ј 2 I1 2Ј ϩ 1 ͓͑ Ϫ ͒ Ϫ ͔2ϩ 1 2 2 M R cos a 2Ј R sin 3 2 I3 3 ϩMga͑1Ϫcos ͒ϭE. ͑6͒ This gives one relation among 1Ј , 2Ј , and 3 . The two nonobvious constants of the motion to which we now turn will give us two more relations for these three compo- nents of .
IV. THE JELLETT CONSTANT Fig. 4. The relevant components of the torque with respect to G are z ϭ ϭ ϵ We notice from Fig. 2 that the torque about G in the r F1ЈaЌ , and 3 F1ЈrЌ , where F1Ј Fx is the component of F normal to ͑ ˆ Ј͒ ϭ ͑ ͒ the plane i.e., the component along 1 . The other components of F, Fy , direction vanishes, i.e., r 0. From 2 we then have • and Fz , do not contribute to z and 3 . ˙ ϭ ͑ ͒ L•r 0. 7 As we shall prove shortly, d a ͩ L Ϫ L ͪ ϭ0, ˙ϭ ͑ ͒ dt z R 3 L•r 0, 8 so that we then have obviously or d a ͑L r͒ϭ0, ͑9͒ L Ϫ L ϭJ ͑a constant͒. ͑13͒ dt • z R 3 ϭ Ϫ ˆ ͑ ͒ i.e., L•r is a constant of the motion, By substituting r Rzˆ a3 in 10 , and dividing both sides by R, we see that ͑10͒ and ͑13͒ are equivalent. Note that this L rϭconst. ͑10͒ • argument does not depend on whether F1Ј is due to static or To establish ͑8͒, we first express r˙ in a convenient form. kinetic friction, so that ͑10͒ is valid whether or not slipping ϭ Ϫ ˆ ˙ϭϪ ˆ˙ occurs. From the definition r Rzˆ a3 we have r a3. But for a The Jellett constant has been rediscovered a number of rotating rigid body, for the body-fixed vector 3ˆ we have 3ˆ˙ times,7,16,30 most recently by Leutwyler30 in 1994 in the form ͑ ͒ ϭϫ3ˆ . Hence we get51 13 . In this form, the Jellett constant J has the dimension of angular momentum, so that the linear combination ͑13͒ of r˙ϭϪaϫ3ˆ . ͑11͒ angular momentum components is a constant of the motion. ϭ ϩ ͑ ͒ ͑ ͒ By using the geometric relation Lz L3 cos L2Ј sin , From 11 and 5 we have ϭ ϭ and the principal axes values L3 I3 3 , L2Ј I1 2Ј , we can ˙ϭϪ ϩ⌬ ˆ ͒ ϫ ˆ ͒ ͑ ͒ rewrite ͑13͒ in the form close to that found by Jellett and L r a͑I1 I 33 ͑ 3 . 12 • • Routh: ϫ ˆ ˆ ϫ ˆ ϫ ˆ The terms ( 3) and 3 ( 3) vanish since ( 3)is ϩ Ϫ ͒ ϭ ͑ ͒ • • I1R sin 2Ј I3͑R cos a 3 const. 14 perpendicular to both and 3ˆ , which establishes ͑8͒. The constant of the motion ͑10͒ is obviously the scalar When 2Ј is expressed in terms of Euler angles, we recover product of L and r, as was first pointed out by O’Brien and the form given by Jellett and Routh. 13 19,31 ͑ ͒ Synge. This constant is not, as sometimes stated, the Equation 14 is the second relation among 1Ј , 2Ј , and ϭ , ͑6͒ being the first. We now find another one. component of L along r, i.e., L•rˆ. Since L•r L•rˆr, where 3 rϵ͉r͉, and r obviously varies during the motion ͑see Fig. 2͒, 34 it is clear that L•rˆ is not a constant of the motion. Jellett V. THE ROUTH CONSTANT first found this constant ͓in a form close to ͑14͒ given below͔ by an approximate argument for the case that there is slip- As we shall see, the Routh constant, like the Jellett con- ping. It was Routh40 who first showed that ͑10͒ is an exact stant, is a kinetic quantity and does not involve the gravita- constant of the motion whether or not there is slipping! tional strength g ͓unlike the total energy ͑6͔͒. Unlike the Since constants of the motion are unusual for dissipative Jellett constant ͑14͒, it is quadratic in the angular velocities systems, we sketch another derivation which brings out more and involves rotational kinetic energy. To derive it, we start clearly why ͑10͒ is also valid if slipping occurs. Examining by eliminating the reaction force F between the two equa- ͑ ͒ ͑ ͒ Fig. 4 we note that only the component F1Ј of F normal to tions of motion 1 and 2 : the plane of the figure contributes to the torque components ˙ ϭ ϭ ϭ LϭϪrϫ͑Mv˙Ϫw͒. ͑15͒ z F1ЈaЌ and 3 F1ЈrЌ . But aЌ a sin and rЌ ϭ ϭ ˙ ˙ ϭ ϫ R sin , so that z / 3 a/R. Hence we have Lz /L3 a/R Inspection of Fig. 2 shows that r w is perpendicular to the from the equation of motion ͑2͒, so that ZO3 plane ͑the plane of the figure͒, and therefore has no
823 Am. J. Phys., Vol. 68, No. 9, September 2000 C. G. Gray and B. G. Nickel 823 ͑ ͒ ˆ I I 3-component. Hence taking the scalar product of 15 with 3 ϭϪ 3 2 Ϫ 3 2Ϫ 2 I3 3 ˙ 3 M r3 3 ˙ 3 M r3r˙ 3 3 MrЌr˙Ќ 3 eliminates w: I1 I1 ˙ ϭ ˙ϫ ˆ ϭ ˆ ϫ ˙ ͑ ͒ Ϫ 2 ͑ ͒ L3 Mv r•3 M3 v•r, 16 MrЌ 3 ˙ 3 . 23 ϫ Equation ͑23͒ can be written where we have used the cyclic property of A B•C. From vϭϫr, we have v˙ϭ˙ ϫrϩϫr˙. Using ͑11͒ and d 1 1 1 I ͫ 2ϩ 2 2ϩ 3 22ͬϭ ͑ ͒ the standard decomposition of Aϫ(BϫC), we find I3 3 MrЌ 3 M r3 3 0. 24 dt 2 2 2 I1 ˆ ϫ ˙ϭ ˙ Ϫ ϩ ϫ ˆ ͑ ͒ 3 v r3 r ˙ 3 a 3 3. 17 From ͑24͒ we find the constant of the motion From ͑17͒ we get 1 1 1 I3 ͫ I ϩ MR2 sin2 ϩ M ͑R cos Ϫa͒2ͬ2 ˆ ϫ ˙ ϭ ͑˙ ͒Ϫ 2 ϩ ͑ ˆ ϫ ͒ ͑ ͒ 2 3 2 2 I 3 3 v•r r3 •r r ˙ 3 aR 3 • 3 zˆ , 18 1 ϭ ͑ ͒ where we have used rϭRzˆϪa3ˆ to obtain the last term in const. 25 ͑18͒. The Routh constant ͑25͒ gives us a third relation among The quantity ˙ r in ͑18͒ can be expressed in terms of the components 1Ј , 2Ј , and 3 . We explore some con- • ͑ ͒ ͑ ͒ simpler quantities using the Jellett condition ͑7͒. We first use sequences later. Equation 25 involves only 3 and .We ͑ ͒ ͑ ͒ 5 to get can bring in 2Ј using the Jellett constant 14 , or using z ϭ ϩ 3 cos 2Ј sin and the Jellett relation, we can rewrite L˙ ϭI ˙ ϩ⌬I˙ 3ˆ ϩ⌬I 3ˆ˙ , ͑19͒ ͑ ͒ 1 3 3 25 in terms of 3 and z . After some manipulation the result takes the form where 3ˆ˙ ϭϫ3ˆ because of the pure rotation of the body- ͑ ͒ ͑ ͒ ͑⌬I͑I ϩMR2͒ϪI Ma2͒2Ϫ2C MR ϩI MR22 fixed unit vectors. Multiplying 19 with •r, and using 7 , 3 3 3 J z 1 z ˙ ϭ ϭ ͑ ͒ L•r 0, thus gives const, 26 ͑˙ ͒ϩ⌬ ϩ⌬ ͑ ˆ ϫ ͒ϭ ͑ ͒ where C is the Jellett constant occurring on the RHS of I1 •r Ir3 ˙ 3 IR 3 • 3 zˆ 0, 20 J ͑14͒. The Routh constant was rediscovered by Isaeva19 in where we have again used rϭRzˆϪa3ˆ to get the last term in 1959 in the form ͑26͒. We do not agree52 with her physical ͑20͒. interpretation of this constant, however. The form ͑25͒ ap- Substituting ˙ r from ͑20͒ into ͑18͒, and the result into pears to be the most useful—see Sec. VI. • ͑ ͒ ͑16͒, and replacing L˙ by I ˙ on the left-hand side ͑LHS͒ The physical significance of the Routh constant 25 or 3 3 3 ͑26͒ is somewhat obscure. It is a kinetic energy type of con- gives stant, and appears to reflect the fact that the constraint force ⌬I ⌬I F is workless, and therefore conserves the total kinetic en- ϭϪ 2 Ϫ ͑ ˆ ϫ ͒Ϫ 2 I3 ˙ 3 M r3 ˙ 3 M Rr3 3 3 zˆ Mr ˙ 3 ergy. It also expresses conservation of the sign of in I1 I1 • 3 rolling motion—see Sec. VI and Ref. 33. ͑iii͒ For the limit- ϩ ͑ ˆ ϫ ͒ ͑ ͒ → ͑ ͒ MaR 3 • 3 zˆ . 21 ing case R 0, the Routh constant 25 can be reduced to We now manipulate ͑21͒ as follows. ͑i͒ We multiply both MaR cos L ϭconstͩ 1ϩ ͪ ϩO͑R2͒, ͑27͒ sides by 3 in order to obtain the energy rate of change 3 ϩ 2 I1 Ma (d/dt) 1 I 2 on the LHS. ͑ii͒ On the right-hand side ͑RHS͒ 2 3 3 and hence L ϭconst for Rϭ0. Substituting ͑27͒ into ͑13͒ we ˆ ϫ ϭϪ 3 we note that 3 zˆ sin uˆ, where uˆ is a unit vector per- find that the Jellett constant in the limit R→0 can be written pendicular to the plane of Fig. 2, pointing outward. We then as note that uˆ ϵ˙ . Replacing r2 in the third term on the RHS • Ma2 cos by r2ϭr2ϩr2 , where r is the component of r along 3ˆ and ϭ ϩ ͑ → ͒ ͑ ͒ 3 Ќ 3 Lz const ϩ 2 L3 R 0 . 28 I1 Ma rЌ the remaining part perpendicular to 3ˆ , and canceling the 52 resulting r2 term against part of the first term gives Using the transformation law for angular momentum under 3 an origin shift we then find that ͑28͒ is equivalent to I I ϭϪ 3 2 Ϫ ͩ 3Ϫ ͪ 2˙ L ϭconst ͑R→0͒, ͑29͒ I3 3 ˙ 3 M r3 3 ˙ 3 M 1 Rr3 3 sin CZ I1 I1 where LC is the angular momentum with respect to the con- Ϫ 2 ϩ 2˙ ͑ ͒ → ϭ MrЌ 3 ˙ 3 MaR 3 sin . 22 tact point C. Since, for R 0, we have LC3 L3 , we can → From the second term on the RHS of ͑22͒, we obtain two write the two conservation laws in this limit (R 0) as ϭ ϭ Ϫ LC3 const and LCZ const. These two conservation laws terms from the two terms in (I3 /I1 1). When we replace r3 46 Ϫ Ϫ are the usual ones for a symmetric top with a fixed point by (R cos a) in the second term, from the 1 we obtain a ͑ ͒ ͑ ͒ term which cancels the last term in ͑22͒. From the relations tip. Equations 13 and 25 thus can be thought of as gen- eralizations from a pointed tip to a rounded one. Alterna- r ϭR cos Ϫa and rЌϭR sin ͑see Fig. 3: rЌ here is the 3 tively, they can be thought of as modifications arising from ͒ ϭϪ ˙ equal to r2Ј there , we easily find r˙ 3 R sin and rЌr˙Ќ the change from a perfectly smooth surface to a perfectly ϭ 2˙ ͑ ͒ R sin cos . Using these relations we find that 22 be- rough one; a perfectly smooth surface conserves L3 and Lz comes ͑see Appendix B͒.
824 Am. J. Phys., Vol. 68, No. 9, September 2000 C. G. Gray and B. G. Nickel 824 VI. DISCUSSION
The three constants of the motion ͑6͒, ͑14͒, and ͑25͒ give us three relations for the angular velocity components 1Ј , 2Ј , and 3 . Thus, during the top motion, for a given value ͑ ͒ ͑ ͒ ͑ ͒ of , 25 fixes 3 . Then 14 fixes 2Ј , and then 6 fixes ϵ˙ the tipping rate 1Ј . Thus we have a functional relation of the form ˙ 2ϭ f (). This is of standard one-dimensional 53 1 2ϭ Ϫ type 2 mx˙ E V(x), and can be formally integrated im- mediately to give t(),tϭ͐d/ͱf (), which can be in- verted to give (t). The integral can be done numerically if needed. A full study would be lengthy since, even for a given top with definite parameters (I1 ,I3 ,a), four initial conditions (0) ˙ (0) (0) (0) must be chosen, , , , 3 , which set the con- 2˙ 2 2Ј Fig. 5. Curves of f 1 vs , for various values of the Jellett constant KJ for ͑ ͒ stants E, KJ , KR , where KJ and KR are the Jellett and Routh our particular tippe top. Only trajectories the horizontal tielines in or near the upper hemisphere (Շ/2) are shown. The end points of the tielines are constants. One can also look for turning points m , defined ϭ the turning points , where ˙ ϭ0. The scale of the K axis is set by point by f ( m) 0. For steady precession at angle p , two turning m J ϭ ͓ 1/2͔ 2˙ 2 ϭ ϭ P where KJ 8.37 in units of (g/R) . The vertical scale is set by f points coincide so that f ( p) 0 and f Ј( p) 0, where f Ј 1 ϭ Љ Ͻ ϭ18.6 ͓in units of MgR͔ at point P; the corresponding energy Eϭ25.1 ͑and df/d . The precession will be stable if f ( p) 0, since ϭ ͒ ϭ correspondingly EЈ 9.29 ensures Fz 0 at the outer turning point. ˙ 2ϭ ¨ ϭ 1 Ј Ӎ 1 Љ Ϫ f ( ) implies 2 f ( ) 2 f ( p)( p) for near p . To illustrate the allowed trajectories qualitatively, we make some plots, for our particular top, of essentially ˙ 2 This can obviously be put in the standard form stated earlier, 2 ϵ f ()vs ͑see below͒; in the one-dimensional case this ˙ ϭ f (). 1 2ϵ Ϫ Figure 5, generated using MAPLE software, shows plots of corresponds to plotting 2 mx˙ E V(x), rather than V(x), 2˙ 2 vs x. We choose convenient basic units as follows: length, f 1 vs for various values of the Jellett constant KJ . The mass, and acceleration are in units of R, M, and g, respec- values of the top parameters for our particular top, a, I1 , and tively. The derived units of time, angular frequency, force, I , are taken from Appendix A. The Routh constant K has 1/2 1/2 3 R energy, and moment of inertia are then (R/g) ,(g/R) , been fixed at K ϭ10 for all curves, corresponding to the 2 R Mg, MgR, and MR , respectively. typical large initial spin rate (0)ϵ(0)/2 of about 20 ͑ ͒ ͑ ͒ 3 3 In our units, versions of the Jellett 14 and Routh 25 cycles/s. The energy E is adjusted for each curve to the maxi- constants are then mum allowed such that the normal force Fz does not become I negative; if allowed to occur, such a negative force would ϩ 3 Ϫ ͒ ϭ ͑ ͒ sin 2Ј ͑cos a 3 KJ , 30 mean in reality that the top would jump off the surface. In I1 55 our units, Fz is given as a function of by f ͑͒22ϭK2 , ͑31͒ 3 3 R ͑͒ϭ ϩ ͑ 1 Ј͑͒ϩ ͑͒͒ ͑ ͒ Fz 1 a 2 sin f cos f . 37 respectively, where For trajectories in the upper hemisphere (р/2), only I ͒2ϭ ϩ 2 ϩ 3 Ϫ ͒2 ͑ ͒ Ј ͑ ͒ f 3͑ I3 sin ͑cos a . 32 f ( )in 37 can become negative; it is largest at the outer I1 ͑ ˙ ϭ ͒ ϭ turning points m points where 0 . The line Fz 0 in the 2 To eliminate the cross term 2Ј 3 and the 3 term from the horizontal KJ , plane of Fig. 5 connects the outer turning energy equation ͑6͒, we form ͑in our units͒ the shifted energy points. The scale of the KJ axis is set by point P, where Јϵ Ϫ 2 ϩ 2 ϭ ͓ ͔ ͑ ͒ 2E 2E KR (I1 /I3)KJ . KJ 8.37, the value for which ¯ in 36 vanishes at ϭ0. At this point the centrifugal barrier f 2/sin2 for →0in 2EЈϭ2a͑1Ϫcos ͒ϩ f ͑͒22 ϩ f ͑͒22 , ͑33͒ 2 2 2Ј 1 1Ј ͑36͒ is nullified by the more rapid (ϳ4) vanishing of the where ͓ ͔ 2 2˙ 2 factor ¯ . The vertical scale is set by the value of f 1 at 2˙ 2ϭ I P, the highest peak, f 1 18.6. ͒2ϭ ϩ 1 2 ϩ Ϫ ͒2 ͑ ͒ f 2͑ I1 sin ͑cos a , 34 As expected, the allowed trajectories in Fig. 5 correspond I3 to a series of tielines in the KJ , plane connecting the inner ͒2ϭ ϩ ϩ 2Ϫ ͑ ͒ f 1͑ I1 1 a 2a cos . 35 and outer turning points. We cannot draw tielines below the K , plane for the reason explained earlier ͑such trajectories ͑ ͒ ͑ ͒ ͑ ͒ J Eliminating 2Ј from 33 using 30 and 31 yields a con- would generate a negative normal force͒, but we can imagine ˙ 2ϵ2 venient form for 1Ј: drawing them above the plane, the trajectories having corre- spondingly lower energy. The limiting cases, the tielines of 2˙ 2ϭ ЈϪ ͑ Ϫ ͒ f 1 2E 2a 1 cos zero length, are the points at the peaks of the curves. These 2 2 points represent trajectories with uniform precession at con- f 2 I3 KR ϭ Љ Ϫ ͫ Ϫ ͑ Ϫ ͒ ͬ ͑ ͒ stant angle p . We can see that for these points f ( p) 2 KJ cos a . 36 sin I1 f 3 Ͻ0, so that the precession is stable.
825 Am. J. Phys., Vol. 68, No. 9, September 2000 C. G. Gray and B. G. Nickel 825 Fig. 7. Physical specifications of our tippe top: the mass Mϭ11.5 g, R ϭ ϭ ϭ ϭ ϭ 2 1.6 cm, c 0.82 cm, l 1.92 cm, r 0.32 cm, a/R 0.070, I3 /MR ϭ 2ϭ ͑ Fig. 6. Top view of Fig. 5, with a new criterion for allowed trajectories. The 0.408, and I1 /MR 0.399. Recall that for a uniform solid sphere I1 ͑ ϭ ϭ 2 energies of the allowed trajectories of Fig. 5 shown as tielines there, and I3 0.4 MR . For the truncated sphere, for calculating I1 and I3 ,the here͒ have been reduced to the maximum allowed such that the top does not origin of the 1,3 axes is displaced to G.͒ ϭ ϭ slip, assuming s 0.5. The line Fh/Fz 0.5 indicates the points on the trajectories ͑not always exactly the outer turning point͒ where the top is on the verge of slipping. The shaded region shows the reduced region of the 1 (0)2 2Mga/ 2 I3 , which, for our particular top, from the data KJ , plane allowed compared to that allowed in Fig. 5. 3 of Appendix A, is in the range 1%–25% for initial spin rates (0) 3 generated by hand. On the other hand, the Routh con- ͑ ͒ (0) In showing the allowed trajectories in Fig. 5 the tielines , stant sets the loss in kinetic energy at about 20% for all 3 . we have assumed the top does not slip anywhere on the In general these conditions disagree, again showing that the trajectory. To see how realistic this assumption is for normal transition ϳ0→ is in general impossible without slip- surfaces, we calculate the ratio Fh /Fz at each point on the ping. ϭͱ 2ϩ 2 trajectory, where Fh Fx Fy is the magnitude of the hori- There are issues that could be explored further. For ex- zontal tangential force at the contact point. If F /F exceeds ample, the precise physical significance of the Routh con- h z 58 the coefficient of static friction , slipping will in fact oc- stant remains elusive. In this connection it might be useful s to try to find a direct connection between this constant of the cur. In our units the components of Fh can be calculated as functions of from56 motion and the underlying symmetries of the system. It would also be of interest to explore the allowed trajectories ͱ ͑ ͒ I3KR f I3 Figs. 5 and 6 for tops of various shapes. For example, for F ͑͒ϭϪ ͩ cos Ϫ ͑cos Ϫa͒ͪ , ͑38͒ ϭϩ x f 3 I our tippe top a/R 0.07, but for an ordinary top a/R might 3 1 be of order Ϫ10. In conclusion, we think the study of the ϩ 1 ЈϪ 2 ϩ constants of the motion for the nonsliding top with a spheri- a sin Fz 2 I1 f I1 cot 2Ј I3 2Ј 3 ͑͒ϭ cal peg is instructive regarding some aspects of the motion of Fy Ϫ , 1 a cos a real top, and also provides a nontrivial example of a system ͑ ͒ 39 which is highly nonlinear yet integrable, one which deserves ͑ ͒ where Fz( ) is given by 37 , and 2Ј( ) and 3( ) are to be better known. ͑ ͒ ͑ ͒ ϭ given by 30 and 31 . As an example we take s 0.5. ACKNOWLEDGMENTS Figure 6 shows the trajectories allowed in the KJ , plane now. It is seen that a considerable fraction of the trajectories The authors thank NSERC ͑Canada͒ for financial support, previously allowed are now unallowed, and the allowed ones Don Sprung, Jim Hunt, and the referees for suggestions, and have a smaller range in . Geoff Lee-Dadswell for assistance with the figures. We can also draw some qualitative conclusions from the individual expressions for the three constants of the motion. APPENDIX A: SPECIFICATIONS OF A COMMON From the Routh constant ͑25͒, for example, because the COMMERCIALLY AVAILABLE TIPPE TOP quantity in brackets does not go to zero or infinity for any value of , we see that 3 cannot change sign during the The top in our possession is made of light hardwood. It is motion. This alone is a proof that sliding friction is essential a truncated sphere of uniform density with a cylindrical for complete tippe top behavior, since it is obvious from Fig. stem, thus forming a symmetric top with principal axes mo- 1 that 3 does change sign. Assuming sliding does occur, ments of inertia I1 , I1 , and I3 . Figure 7 gives the specifica- Jellett57 ͑see also Gallop3 and Leutwyler30͒ shows from the tions. In particular, the asymmetry parameters ⑀ϵa/R and Jellett constant alone ͑which we have seen is valid even with ␦ϵ⌬ ⑀ϭ ␦ϭ I/I1 have the values 0.070 and 0.0226. sliding͒ that the tippe top does indeed tip up to the vertical ϭ ( ) under certain conditions. Another simple conse- APPENDIX B: CONSTANTS OF THE MOTION FOR quence can be found by comparing the implications of the A PERFECTLY SMOOTH SURFACE energy and Routh constants for the situation illustrated in Fig. 1, i.e., changing from near zero to . With no fric- This case has also been discussed by Routh59 and is much tional dissipation the fractional kinetic energy loss is simpler than the case of a perfectly rough surface. The con-
826 Am. J. Phys., Vol. 68, No. 9, September 2000 C. G. Gray and B. G. Nickel 826 13 straint force F now has only a vertical or normal component 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. Fz . We rewrite the equations of motion 1 and 2 for this 56, 23–35 ͑1954͒. case as 14A. R. Del Campo, ‘‘Tippe Top ͑Topsy-Turvee Top͒ Continued,’’ Am. J. Phys. 23, 544–545 ͑1955͒. Mv˙ϭNϩw, L˙ ϭϪrϫN, ͑B1͒ 15I. M. Freeman, ‘‘The Tippe Top Again,’’ Am. J. Phys. 24,178͑1956͒. 16D. G. Parkyn, ‘‘The Inverting Top,’’ Math. Gazette 40, 260–265 ͑1956͒. ϭ ͑ ͒ ͑ ͒ 17 where N Fzzˆ. The constraint 3 still holds, but 4 does W. Schallreuter, ‘‘Der Spielkreisel mit Selbstaufrichtung,’’ Wiss. Z. der not. Ernst Moritz Arndt-Universita¨t Greifswald 8, 43–46 ͑1958–9͒. There are six coordinates38 and one holonomic constraint 18D. G. Parkyn, ‘‘The Rising of Tops with Rounded Pegs,’’ Physica ͑Am- zϭRϪa cos , where z is the Z component of r, yielding sterdam͒ 24, 313–330 ͑1958͒. 19L. S. Isaeva, ‘‘On the Sufficient Conditions of Stability of Rotation of a five degrees of freedom. There are five constants of the mo- Tippe-Top on a Perfectly Rough Horizontal Surface,’’ J. Appl. Math. tion, as we show, so that the system is integrable. Mech. 23, 403–406 ͑1959͒. From the absence of forces in the X and Y directions, we 20J. B. Hart, ‘‘Angular Momentum and Tippe Top,’’ Am. J. Phys. 27,189 ϭ ͑ ͒ have the momentum constants Mvx const and Mvy 1959 . ϭ 21F. Johnson, ‘‘The Tippy Top,’’ Am. J. Phys. 28, 406–407 ͑1960͒. const. From the absence of torques around the 3 and Z 22 ϭ G. D. Freier, ‘‘The Tippy-Top,’’ Phys. Teach. 5, 36–38 ͑1967͒. axes, we have the angular momentum constants L3 const 23For a delightful picture of Bohr and Pauli playing with a tippe top, see S. ϭ and Lz const. From the absence of dissipation, we have the Rozental, ed., Niels Bohr: His Life and Work as Seen by His Friends and energy Eϭconst, where Colleagues ͑Interscience, New York, 1967͒,p.209. 24J. C. Lauffenburger, ‘‘A Large-Scale Demonstration of the Tippe Top,’’ 2 2 ϭ 1 ϩ 1 ϩ 1 2ϩ 1 2ϩ 1 2ϩ 1 2 Am. J. Phys. 40, 1338 ͑1972͒. E 2 I1 1Ј 2 I1 2Ј 2 I3 3 2 Mvx 2 Mvy 2 Mvz 25R. J. Cohen, ‘‘The Tippe Top Revisited,’’ Am. J. Phys. 45, 12–17 ͑1977͒. ϩMga͑1Ϫcos ͒. ͑B2͒ 26J. Walker, The Flying Circus of Physics ͑Wiley, New York, 1977͒,p.43. 27T. R. Kane and D. A. Levinson, ‘‘A Realistic Solution of the Symmetric The third, fourth, and fifth terms are constants, so that ͑B2͒ Top Problem,’’ J. Appl. Mech. 45, 903–909 ͑1978͒. simplifies to 28N. G. Chataev, Theoretical Mechanics ͑MIR, Moscow, 1987; revised edi- tion by Springer, Berlin, 1989͒, p. 178. Chataev discusses the very similar 1 2 ϩ 1 2 ϩ 1 2ϩ ͑ Ϫ ͒ϭ 2 I1 1Ј 2 I1 2Ј 2 Mvz Mga 1 cos const. Chinese top. ͑B3͒ 29V. D. Barger and M. G. Olsson, Classical Mechanics: A Modern Perspec- tive ͑McGraw–Hill, New York, 1995͒, 2nd ed., p. 273. ϭ ϭ ˙ ϵ˙ 30 ͑ ͒ Here vz z˙ a sin , and 1Ј . Using H. Leutwyler, ‘‘Why Some Tops Tip,’’ Eur. J. Phys. 15, 59–61 1994 . 31A. Gray, A Treatise on Gyrostatics and Rotational Motion ͑MacMillan, ϭ ϩ ͑ ͒ Lz I3 3 cos I1 2Ј sin , B4 London, 1918, reprinted by Dover, 1959͒. 32 ͑ R. H. Deimel, Mechanics of the Gyroscope MacMillan, New York, 1929; we can eliminate 2Ј also, and obtain, as before, a one- reprinted by Dover, 1950͒. dimensional equation of standard type ˙ 2ϭ f (). 33Some qualitative arguments may be helpful here. ͑i͒ Friction of some type is necessary for the top to make the transition shown in Fig. 1, since z is ͒ reduced ͑conservation of energy—the center of mass has risen͒, so that a z a Electronic mail: [email protected] b͒ torque, which can only arise from friction, must have played a role. To see Electronic mail: [email protected] ͑ ͒ 1The history of the theory of motion of solid bodies on a plane surface is that some sliding friction is essential, note ii conservative systems in sketched by Routh ͑Ref. 40, Pt. 2, p. 186͒ for the cases of no friction, and bounded motions generally oscillate between turning points, and do not in ͑ ͒ ͑ ͒ perfect friction ͑i.e., no slipping͒. For the case of sliding friction, accord- general reach a final steady state as observed in Fig. 1 b ,and iii a pure ͑ ͒ ing to Perry ͑Ref. 2, p. 39͒ and Gray ͑Ref. 31, p. 393͒, the earliest work is rolling motion with pivoting allowed but no sliding would produce in ͑ ͒ due to A. Smith and Kelvin in the 1840s. The work of Gallop ͑Ref. 3͒ and Fig. 1 b the top with angular momentum pointing down, rather than still ͑ ͒ Jellett ͑Ref. 34͒ is also seminal. up as in Fig. 1 b . This can be seen by visualizing the top motion, or by ͓ ͑ ͒ 2J. Perry, Spinning Tops and Gyroscopic Motions ͑Sheldon, London, 1890, slowly rolling an actual top. The modern way of stating ii is ‘‘Conser- reprinted by Dover, 1957͒,p.43. vative Systems Have No Attractors,’’ see, e.g., R. L. Borrelli and C. S. ͑ ͒ ͔ 3E. G. Gallop, ‘‘On the Rise of a Spinning Top,’’ Trans. Cambridge Philos. Coleman, Differential Equations Wiley, New York, 1998 ,p.457. 34 ͑ Soc. 19, 356–373 ͑1904͒. Some of Gallop’s results are summarized in J. H. Jellett, A Treatise on the Theory of Friction MacMillan, London, ͒ Deimel ͑Ref. 32, p. 93͒ and Gray ͑Ref. 31, p. 396͒. 1872 , p. 185. 35 4H. Crabtree, An Elementary Treatment of the Theory of Spinning Tops and We consider only sliding kinetic friction here, the most important type for ͑ Gyroscopic Motion ͑Longmans Green, London, 1909, reprinted by the tippe top. For tops in general, there is also rolling friction due to finite ͒ ͑ ͒ ͑ Chelsea, 1967͒,p.5. elasticity of the surfaces in contact , boring or pivoting friction due to ͒ 5A. D. Fokker, ‘‘The Tracks of Tops Pegs on the Floor,’’ Physica ͑Amster- finite sized area of contact , and air friction. 36 ͑ dam͒ 18, 497–502 ͑1952͒, see also ‘‘The Rising Top, Experimental Evi- M. Tabor, Chaos and Integrability in Nonlinear Dynamics Wiley, New ͒ dence and Theory,’’ 8, 591–596 ͑1941͒. York, 1989 . 37 6C. M. Braams, ‘‘On the Influence of Friction on the Motion of a Top,’’ V. I. Arnold, Mathematical Methods of Classical Mechanics ͑Springer, Physica ͑Amsterdam͒ 18, 503–514 ͑1952͒; ‘‘The Symmetrical Spherical New York, 1989͒, 2nd ed. 38 Top,’’ Nature ͑London͒ 170,31͑1952͒; ‘‘The Tippe Top,’’ Am. J. Phys. We say local because the tippe top has five global degrees of freedom 22,568͑1954͒. (xG ,y G , , , ) corresponding to the X,Y coordinates of the center of 7 ͑ ϭ Ϫ ͒ N. M. Hugenholtz, ‘‘On Tops Rising by Friction,’’ Physica ͑Amsterdam͒ mass G the Z coordinate is constrained by zG R a cos , and the three 18, 515–527 ͑1952͒. Euler angles giving the orientation. There are two independent nonholo- 8J. A. Haringx, ‘‘De Wondertol,’’ De Ingenieur 4, 13–17 ͑1952͒. nomic ͑rolling͒, or local, constraints, thus giving three local degrees of 9J. A. Jacobs, ‘‘Note on the Behaviour of a Certain Symmetrical Top,’’ freedom. Am. J. Phys. 20, 517–518 ͑1952͒. 39We have found no references to Routh’s discussion of the integrability of 10D. Van Ostenburg and C. Kikuchi, ‘‘Some Analogies of the Tippe Top to the nonsliding tippe top. Even a recent monograph devoted to integrable Electrons and Nuclei,’’ Am. J. Phys. 21, 574 ͑1953͒. top systems does not mention it: M. Audin, Spinning Tops: A Course on 11F. Schuh, ‘‘Beweging van een Excentrisch Bezaarde Bol Over een Hori- Integrable Systems ͑Cambridge U. P., New York, 1996͒. zontaal Vlak in Verband met de Tovertol ‘Tippe Top,’ ’’ Proc. K. Ned. 40E. J. Routh, A Treatise on the Dynamics of a System of Rigid Bodies, Pt. Akad. Wet., Ser. A: Math. Sci. 56, 423–452 ͑1953͒. 1, The Elementary Part ͑MacMillan, London, 1860, 1905͒, 6th ed., Pt. 2, 12W. A. Pliskin, ‘‘The Tippe Top ͑Topsy-Turvy Top͒,’’ Am. J. Phys. 22, The Advanced Part ͑reprinted by Dover, New York, in 1960 and 1955, 28–32 ͑1954͒. respectively͒. See especially Pt. 2, p. 192.
827 Am. J. Phys., Vol. 68, No. 9, September 2000 C. G. Gray and B. G. Nickel 827 41 ͑ ͒ ͑ ϫ ͔ ͓ → Edward John Routh 1831–1907 is remembered for the Routhian Ref. r˙C , and do not vanish. An exception occurs in the limit R 0—see the ͒ ͑ ͒ ͑ ͒ ͔ 42 , the Routh–Hurwitz and other stability criteria Ref. 43 , and the discussion below Eq. 29 . Here C is the torque about C,andr˙C the Routh rules ͑Ref. 44͒ for moments of inertia, as well as for his books ͑Ref. velocity of the contact point as traced out on the horizontal surface ͑see 40͒. Interestingly, in the Cambridge Mathematical Tripos examination of Ref. 51͒. ͑ ͒ 1854 Routh finished first senior wrangler , and Maxwell second. His 53See, for example, Osgood ͑Ref. 45, p. 456͒, Synge and Griffith ͑Ref. 44, ͑ ͒ books have been out of fashion for some time Ref. 45 , but are an invalu- pp. 328, 388͒ Goldstein ͑Ref. 42, pp. 75, 76, 215, 216, or Chataev ͑Ref. able source of results and inspiration. 28, pp. 130, 136, 210͒. 42 ͑ H. Goldstein, Classical Mechanics Addison–Wesley, Cambridge, MA, 54D. J. McGill and J. G. Papastavridis, ‘‘Comments on ‘Comments on Fixed ͒ 1980 , 2nd ed., p. 352. Points in Torque-Angular Momentum Relations,’ ’’ Am. J. Phys. 55,470– 43D. R. Merkin, Introduction to the Theory of Stability ͑Springer, New York, 471 ͑1987͒. 1997͒, pp. 84, 111. 55The normal force is given by ͓see Fig. 2 or Eq. ͑1͔͒ F ϭMgϩMv˙ . Since 44J. L. Synge and B. A. Griffith, Principles of Mechanics ͑McGraw–Hill, z z ϭ ϭ Ϫ ϭ ¨ϩ ˙2 ˙ 2 New York, 1959͒, 2nd ed., p. 293. v˙ z z¨ and z R a cos , we have v˙ z a sin a cos . From 45 ϭ ¨ ϭ According to William Fogg Osgood, author of a number of textbooks, f ( ) we obtain f Ј( )/2, so that Fz as a function of is Fz( ) including Mechanics ͑MacMillan, London, 1937; reprinted by Dover, 1 ϭMgϩMa( sin fЈ()ϩcos f()). 1965͒, ‘‘Routh’s exposition of the theory is execrable, but his lists of 2 56We choose X,Y axes as in Fig. 4. F ϵF can be obtained from ͓see the problems, garnered from the old Cambridge Tripos papers, are capital.’’ x 1Ј ͑ ͔͒ ϭ ϭ ͑ ͒ ϭ ͑Osgood, p. 246͒. argument above Eq. 13 FxR sin 3 I3 ˙ 3 . From 31 we get ˙ 3 46 Ϫ ˙ 2 ˙ ͑ ͒ ͑ ͒ R. Baierlein, Newtonian Dynamics ͑McGraw–Hill, New York, 1983͒, KR f 3 / f 3. Evaluating f 3 from 32 then yields expression 38 for Fx Chap. 7. ϭ ˙ ˆ Јϭ ˙ ˆ Ј given in the text. To obtain Fy we start with L,sothat •1 L•1 47E. A. Milne, Vectorial Mechanics ͑Methuen, London, 1948͒, Chaps. 15– ϭ ˆ Ј Ϫ ˆ˙ Јϭ Ϫ ˆ˙ Ј ͑ ͒ ˆ˙ Јϭ˙ 17. (d/dt)(L•1 ) L•1 I1 ˙ 1Ј L•1 . Using see Fig. 4 1 yˆ 48 ϭ ϭ ˆЈϪ ˆ ˆ Јϭ Ϫ W. Case, ‘‘The Gyroscope: An Elementary Discussion of a Child’s Toy,’’ ( 2Ј/sin )yˆ ( 2Ј/sin ) (cos 2 sin 3), and •1 Fy(R a cos ) ͑ ͒ Ϫ Ϫ Ϫ ϭ Ϫ 2 Am. J. Phys. 45, 1107–1109 1977 ; W. B. Case and M. A. Shay, ‘‘On the Fza sin ,wegetFy(R a cos ) Fza sin I1 ˙ 1Ј I1 cot 2Ј Interesting Behaviour of a Gimbal-mounted Gyroscope,’’ ibid. 60, 503– ϩI .Wethenuse˙ ϵ¨ ϭ f Ј()/2 to get expression ͑39͒ for F ͑ ͒ 3 2Ј 3 1Ј y 506 1992 . given in the text. 49 The three Euler angles are , with describing the azimuth position 57The full argument ͑Ref. 34͒ is somewhat lengthy, but the essence is as around Z, and the spin position around 3. Since there are X, Y and 1, 2 ͓ ͑ ͔͒ ˙ ϭ ˙ rotational symmetries in the problem, and will contain arbitrary ref- follows. Since see 13 Lz (a/R)L3 , any change in the initial Lz is only ͑ϳ ͒ erence values. a small fraction (a/R) 7% for our top of the change in L3 . Suppose the 50 ͑ ͒ Ϸ Ϸ For a masterful account of nonholonomic constraints, see J. G. Papastavri- top starts as in Fig. 1 a , with 0 and Lz L3 both large. If after some dis, Tensor Calculus and Analytical Dynamics ͑CRC Press, Boca Raton, time the sliding friction has reduced L3 to a small value, say zero, Lz is FL, 1999͒. thus reduced to about 93% of its initial value. This means must have 51 Þ ϭ Ϫ Note that r˙ v. The correct relation is r˙ v r˙C , where rC is the vector increased to about 90°, otherwise Lz would project a large component from the origin of an arbitrary set of space fixed axes to the moving point along the 3 axis. In other words, the center of mass has risen. ϭ ϭϫ 58 C. Using r˙C vO Rzˆ, since point C is always directly under point O A hint is perhaps contained in a result of Gallop ͑Ref. 3͓͒see also Deimel ͑see Fig. 2͒,andvϭϫr, we again get ͑11͒. ͑Ref. 32͒ and Leutwyler ͑Ref. 30͔͒. For the sliding case, where vϭϫr 52Isaeva ͑Ref. 19͒ states ‘‘The projection of the angular momentum about no longer holds ͑see Appendix B͒, Gallop shows that the minimum of the the point of contact on a principal axis of inertia is a constant.’’ We are ͑ ͒ (J) total kinetic energy, at fixed and fixed Jellett constant J 13 ,isKmin unable to confirm this claim. The angular momentum about contact point ϫ()ϭJ2/2I (), where I ()ϭI sin2 ϩI (cos Ϫ⑀)2, with ⑀ϭa/R,is ϵ ϭ ϩ ϫ eff eff 1 3 C, LC , is related to that about G, LG L,byLC L Mr v. Because the effective moment of inertia occurring in Routh’s constant ͑25͒ when L rϭL r, we have L rϭconst as another version of the Jellet constant C• • C• rewritten as ͓(1/2)(I I /MR2)ϩ(1/2)I ()͔2ϭconst. The minimum oc- ͑10͒. This gives L Ϫ(a/R)L ϭconst, where L ϭL cos 3 1 eff 3 CZ C3 CZ C3 curs with the top rotating about r, and the corresponding moment of iner- ϩL sin. Hence L and L are related. However, the individual C2Ј C3 C2Ј tia I is related to I ()byI ()ϭ(r/R)2 I . Whether this has any signifi- ˙ ˙ ˙ ˙ ͓ rˆ eff eff rˆ quantities LC1Ј , LC2Ј , and LC3 can be calculated from LC using an cance for the rolling case is not clear. ͑ ͒ ͑ ͒ ˙ ϭ ϩ 59 ͑ ͒ equation of motion Ref. 54 which differs from 2 , i.e., LC C Mv Routh Ref. 40 , p. 194.
828 Am. J. Phys., Vol. 68, No. 9, September 2000 C. G. Gray and B. G. Nickel 828