Constants of the motion for nonslipping tippe tops and other tops with round pegs

Abstract

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.

Towards a Prototype of a Spherical Tippe Top

“…Since it was established that sliding friction was necessary to explain the Tippe Top inversion [5,9,14], many studies have been dedicated to the analysis of models for tippe tops, involving linear stability analysis of the relative equilibria, numerical simulations, etc. Some studies have addressed the occurrence of transitions between rolling and sliding during the motion, see [13,15,18]. In this paper the presented mathematical results mainly reproduce those in [8,6,7,10,20,16] but our approach is inspired by the hands-on numerical approach as first attempted by Cohen in [9].…”

“…when the non-holonomic constraint V Q = 0 is satisfied) allows for complete reduction of the equations of motion to a second order ode. See [13] for a discussion of this approach. In the pure rolling regime the system is not anymore dissipative and admits three conserved quantities: the energy, E, the Jellett J as before and the Routhian, Routh, given by [13] …”

“…Note further that the constraint V Q = 0 gives U (θ), V (θ) from (3.5). In this approach, as mentioned in [13] one has to check whether a found solution is physically possible, that is, one has to take into account that rolling cannot be sustained if |R t | < µ s R n , where µ s is the coefficient of static friction. In [13] it is remarked that only a few pure rolling precessional solutions satisfy this condition.…”

“…In this approach, as mentioned in [13] one has to check whether a found solution is physically possible, that is, one has to take into account that rolling cannot be sustained if |R t | < µ s R n , where µ s is the coefficient of static friction. In [13] it is remarked that only a few pure rolling precessional solutions satisfy this condition. The analysis however leaves open the possibility to have pure rolling periodic motions around the intermediate states as we discuss below.…”

“…1.1 was introduced in Denmark, several theoretical articles have been published since then, see e.g. [9,13,18,3,21] for a survey of the literature. Since it was established that sliding friction was necessary to explain the Tippe Top inversion [5,9,14], many studies have been dedicated to the analysis of models for tippe tops, involving linear stability analysis of the relative equilibria, numerical simulations, etc.…”