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 axis3 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. The Tippe Top (TT) is known for its counterintuitive behaviour; when it is spun, its rotation axis turns upside down and its center of mass rises. Its behaviour is presently understood through analysis of stability of its straight and inverted spinning solution. The energy of TT is a monotonously decreasing function of time and it appears that it is a suitable Lyapunov function for showing instability of the straight spinning solution and stability of the inverted spinning solution [1,7,10,11,19]. The energy is also a suitable LaSalle function to conclude that for sufficiently large angular momentum L directed close to the verticalẑ-direction the inverted spinning solution is asymptotically attracting [1,19].Since the 1950s [9] it is also known how TT has to be built for the inversion to take place, more precisely that if 0 < α < 1 denotes the eccentricity of the center of mass then the quotient of moments of inertia γ =has to satisfy the inequality 1−α < γ < 1+α. It is also understood that the gliding friction is necessary for converting rotational energy to potential energy.When it comes to describing the actual dynamics of TT, there is very little known. In several papers [5,7,16,22] numerical results of how the Euler angles (θ(t), ϕ(t), ψ(t)) change during inversion are presented, but an understanding of qualitative features of solutions (as well as details of physical forces and torques acting during the inversion) remains a rather unexplored field.There is also an alternative approach [18,21] through the main equation of the TT that shows the source of oscillatory behaviour of θ(t) and of the remaining variables ϕ(t), ψ(t). The main difficulty in making further progress lies in the complexity of the TTs equations of motion.The TT is described, in the simplest setting, by six nonlinear dynamical equations for six variables (θ(t),θ(t),φ(t), ω 3 (t), ν x (t), ν y (t)). Analysis is slightly simplified by the fact that these equations admit one integral of motion, the Jellett integral, and that the energy function is monotonously decreasing in time. To make further progress in reading off the dynamical conte...

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We study the relationship between numerical solutions for inverting Tippe Top and the structure of the dynamical equations. The numerical solutions confirm oscillatory behaviour of the inclination angle θ(t) for the symmetry axis of the Tippe Top. They also reveal further fine features of the dynamics of inverting solutions defining the time of inversion. These features are partially understood on the basis of the underlying dynamical equations.