Dynamics of an Inverting Tippe Top

TL;DR

This paper analytically demonstrates the oscillatory behavior of an inverting tippe top by analyzing its main equation and effective potential, providing insights into the inversion dynamics beyond previous numerical and perturbation studies.

Contribution

It offers a rigorous analytical proof of oscillatory motion during tippe top inversion using the main equation and effective potential analysis.

Findings

Proves the effective potential has a single minimum that moves during inversion.

Shows the inclination angle oscillates during inversion under certain conditions.

Provides estimates for the maximum oscillation period of the inclination angle.

Abstract

The existing results about inversion of a tippe top (TT) establish stability of asymptotic solutions and prove inversion by using the LaSalle theorem. Dynamical behaviour of inverting solutions has only been explored numerically and with the use of certain perturbation techniques. The aim of this paper is to provide analytical arguments showing oscillatory behaviour of TT through the use of the main equation for the TT. The main equation describes time evolution of the inclination angle $\theta(t)$ within an effective potential $V(\cos\theta,D(t),\lambda)$ that is deforming during the inversion. We prove here that $V(\cos\theta,D(t),\lambda)$ has only one minimum which (if Jellett's integral is above a threshold value $\lambda>\lambda_{\text{thres}}=\frac{\sqrt{mgR^3I_3\alpha}(1+\alpha)^2}{\sqrt{1+\alpha-\gamma}}$ and $1-\alpha^2<\gamma=\frac{I_1}{I_3}<1$ holds) moves during the inversion from a neighbourhood of $\theta=0$ to a neighbourhood of $\theta=\pi$. This allows us to conclude that $\theta(t)$ is an oscillatory function. Estimates for a maximal value of the oscillation period of $\theta(t)$ are given.