TIPPE TOP Consider a Spherical Top of Radius R, Except That Its Center of Mass Is Not at the Center of the Sphere, but Rather At

TIPPE TOP YOSI AVRON Abstract. Tippe-top is a top a spherical top that flips on its head when spun fast enough. The reason for that is hidden in an interesting constant of motion. This implies that the top has less energy when standing on its head. Consider a spherical top of radius R, except that its center of mass is not at the center of the sphere, but rather at ~® away from the center of the sphere. The top is moving on top of a table and the normal to the table isz ˆ. If we assume that the sphere touches the table at a point, then the problem has an interesting constant of motion, namely, with L~ the angular momentum of the top about its center of mass: C = L~ ¢ (Rzˆ + ®) Indeed ˙ C˙ = L~ ¢ (Rzˆ + ~®) + L~ ¢ ®˙ Now, with F~ the force at the contact point, the equations of motion is ˙ L~ = (Rzˆ + ~®) £ F From this follows that the first term drops and so C˙ = L~ ¢ ®˙ Now, compute the right hand side in the frame of the top. In this frane ~® is aa fixed vector so ~®˙ = ~! £ ® is perpendicular to both ! and ®. However, since the top is symmetric L~ = I2 ~! + (I3 ¡ I2)(! ¢ ®ˆ)® ˆ L~ is in the plane of ! and ®. Hence C˙ = 0. From this conservation law it follows the tippe top has less energy when its center of mass is above the center of the sphere than when it is below it. Indeed, from the coservation law !a(R + ®) = !b(R ¡ ®) 2 or !a < !b. The kinetic energy is proportional to ! and the result follows. So, if the energy is taken out of the top (for example, by slipping), the top will be forced to flip. Remarkably, gravity played no explicit role in the argument. with gravity present, the argument still holds provided the top is spun fast enough so that the gain in potential energy when the top rises is dominated by the loss of kinetic energy. Date: October 11, 2003. 1.

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Properties of Numerical Solutions Versus the Dynamical Equations
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