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 3ˆ 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 Chaplygin separation equation for a rolling axisymmetric ball has an algebraic expression for the effective potential V(z = cos θ, D, λ) which is difficult to analyse. We simplify this expression for the potential and find a 2-parameter family for when the potential becomes a rational function of z = cos θ. Then this separation equation becomes similar to the separation equation for the heavy symmetric top. For nutational solutions of a rolling sphere, we study a high frequency ω3-dependence of the width of the nutational band, the depth of motion above V(z_min, D, λ) and the ω3-dependence of nutational frequency . These results have bearing for understanding the inverting motion of the Tippe Top, modeled by a rolling and gliding axisymmetric sphere.

We study an equivalent integrated form of the Tippe Top (TT) equations that leads to the Main Equation for the Tippe Top (METT), an equation describing time evolution of the inclination angle θ(t) of inverting TT. We study how the effective potential V(cos θ, D, λ) in METT deforms as TT is inverting and show that its minimum moves from a neighborhood of θ = 0 to a neighborhood of θ = π. We analyse behaviour of θ(t) and show that it oscillates and moves toward θ = π when the physical parameters of the TT satisfy 1 − α² < γ < 1 and the initial conditions are such that Jellett’s integral satisfy

. Estimates for maximal value of the oscillation period of θ(t) are given.