
SIGMA 8 (2012), 019, 22 pages arXiv:1204.1123
http://dx.doi.org/10.3842/SIGMA.2012.019
Tippe Top Equations and Equations for the Related Mechanical Systems
Nils Rutstam
Department of Mathematics, Linköping University, Linköping, Sweden
Received October 21, 2011, in final form March 27, 2012; Published online April 05, 2012
Abstract
The equations of motion for the rolling and gliding Tippe Top (TT) are nonintegrable and difficult to analyze. The only existing arguments about TT inversion are based on analysis of stability of asymptotic solutions and the LaSalle type theorem. They do not explain the dynamics of inversion.
To approach this problem we review and analyze here the equations of motion for the rolling and gliding TT in three equivalent forms, each one providing different bits of information about motion of TT. They lead to the main equation for the TT, which describes well the oscillatory character of motion of the symmetry axis $\mathbf{\hat{3}}$ during the inversion.
We show also that the equations of motion of TT give rise to equations of motion for two other simpler mechanical systems: the gliding heavy symmetric top and the gliding eccentric cylinder. These systems can be of aid in understanding the dynamics of the inverting TT.
Key words:
tippe top; rigid body; nonholonomic mechanics; integrals of motion; stability; gliding friction.
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References
 BouRabee N.M., Marsden J.E., Romero L.A., Tippe top inversion as a
dissipationinduced instability, SIAM J. Appl. Dyn. Syst. 3
(2004), 352377.
 Chaplygin S.A., On a ball's rolling on a horizontal plane, Regul.
Chaotic Dyn. 7 (2002), 131148.
 Chaplygin S.A., On a motion of a heavy body of revolution on a horizontal
plane, Regul. Chaotic Dyn. 7 (2002), 119130.
 Ciocci M.C., Langerock B., Dynamics of the tippe top via Routhian reduction,
Regul. Chaotic Dyn. 12 (2007), 602614,
arXiv:0704.1221.
 Cohen R.J., The tippe top revisited, Amer. J. Phys. 45
(1977), 1217.
 Del Campo A.R., Tippe top (topsyturnee top) continued, Amer. J. Phys.
23 (1955), 544545.
 Ebenfeld S., Scheck F., A new analysis of the tippe top: asymptotic states and
Liapunov stability, Ann. Physics 243 (1995), 195217,
chaodyn/9501008.
 Glad S.T., Petersson D., RauchWojciechowski S.,
Phase space of rolling solutions of the tippe top,
SIGMA 3 (2007), 041, 14 pages,
nlin.SI/0703016.
 Hugenholtz N.M., On tops rising by friction, Physica 18
(1952), 515527.
 Karapetyan A.V., Global qualitative analysis of tippe top dynamics,
Mech. Sol. 43 (1995), 342348.
 Karapetyan A.V., Qualitative investigation of the dynamics of a top on a plane
with friction, J. Appl. Math. Mech. 55 (1991), 563565.
 Karapetyan A.V., Kuleshov A.S., Steady motions of nonholonomic systems,
Regul. Chaotic Dyn. 7 (2002), 81117.
 LaSalle J.P., Some extensions of Liapunov's second method, IRE
Trans. 7 (1960), 520527.
 Moffatt H.K., Shimomura Y., Classical dynamics: spinning eggs  a paradox
resolved, Nature 416 (2002), 385386.
 Moffatt H.K., Shimomura Y., Branicki M., Dynamics of an axisymmetric body
spinning on a horizontal surface. I. Stability and the gyroscopic
approximation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.
460 (2004), 36433672.
 Or A.C., The dynamics of a tippe top, SIAM J. Appl. Math. 54
(1994), 597609.
 Pliskin W.A., The tippe top (topsyturvy top), Amer. J. Phys.
22 (1954), 2832.
 RauchWojciechowski S., What does it mean to explain the rising of the tippe
top?, Regul. Chaotic Dyn. 13 (2008), 316331.
 RauchWojciechowski S., Sköldstam M., Glad T., Mathematical analysis of the
tippe top, Regul. Chaotic Dyn. 10 (2005), 333362.
 Routh E.J., The advanced part of a treatise on the dynamics of a system of
rigid bodies. Being part II of a treatise on the whole subject, 6th ed., Dover Publications Inc., New York, 1955.
 Rutstam N., Study of equations for tippe top and related rigid bodies,
Linköping Studies in Science and Technology, Theses No. 1106,
Matematiska Institutionen, Linköpings Universitet, 2010, available at http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva60835.
 Ueda T., Sasaki K., Watanabe S., Motion of the tippe top: gyroscopic balance
condition and stability, SIAM J. Appl. Dyn. Syst. 4 (2005),
11591194, physics/0507198.
 Zobova A.A., Karapetyan A.V., Analysis of the steady motions of the tippe top,
J. Appl. Math. Mech. 73 (2009), 623630.

