
SIGMA 3 (2007), 041, 14 pages nlin.SI/0703016
http://dx.doi.org/10.3842/SIGMA.2007.041
Contribution to the Vadim Kuznetsov Memorial Issue
Phase Space of Rolling Solutions of the Tippe Top
S. Torkel Glad ^{a}, Daniel Petersson ^{a} and Stefan RauchWojciechowski ^{b}
^{a)} Dept. of Electrical Engineering, Linköpings Universitet
SE581 83 Linköping, Sweden
^{b)} Department of Mathematics, Linköpings Universitet, SE581 83 Linköping, Sweden
Received September 15, 2006, in final form February 05, 2007; Published online March 09, 2007
Abstract
Equations of motion of an axially symmetric sphere rolling and sliding
on a plane are usually taken as model of the tippe top.
We study these equations in the nonsliding regime both in the vector
notation and in the Euler angle variables when they admit
three integrals of motion that are linear and quadratic in momenta.
In the Euler angle variables (θ,φ,ψ) these integrals give separation
equations
that have the same structure as the equations
of the Lagrange top. It makes it possible to describe the whole
space of solutions by representing them in the space of parameters
(D,λ,E) being constant values of the integrals of motion.
Key words:
nonholonomic dynamics; rigid body; rolling sphere; tippe top; integrals of motion.
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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 motion of heavy rigid body of
revolution on horizontal plane, Proc. of the Physical Sciences,
Section of the Society of Amateurs of Natural Sciences 9 (1897), no. 1,
1016.
 Chaplygin S.A.,
On a ball's rolling on a horizontal plane,
Regul. Chaotic Dyn. 7 (2002), 131148.
 Cohen R.J.,
The tippe top revisited,
Amer. J. Phys. 45 (1977), 1217.
 Ebenfeld S., Scheck F.,
A new analysis of the tippe top: asymptotic states and Liapunov stability,
Ann. Physics 243 (1995), 195217, chaodyn/9501008.
 Gray C.G., Nickel B.G,
Constants of motion for nonslipping tippe tops and other tops with rounded pegs,
Amer. J. Phys. 68 (1999), 821828.
 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.,
On the specific character of the application of Routh's theory to systems with differential constraints,
J. Appl. Math. Mech. 58 (1994), 387392.
 Kuleshov A.S.,
On the generalized Chaplygin integral,
Regul. Chaotic Dyn. 6 (2001), 227232.
 Landau L.D., Lifshitz E.M.,
Mechanics, Pergamon Press Ltd, Oxford, 1976.
 Moshchuk N.K.,
Qualitative analysis of the motion of a rigid body of revolution on an absolutely rough plane,
J. Appl. Math. Mech. 52 (1988), 159165.
 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, Dover Publications, New
York, 1905, 131165.
 Zobova A.A., Karapetyan A.V.,
Construction of PoincaréChetaev and Smale bifurcation
diagrams for conservative nonholonomic systems with symmetry,
J. Appl. Math. Mech. 69 (2005), 183194.

