Tippe Top Equations and Equations for the Related Mechanical Systems

TL;DR

This paper analyzes the complex equations governing the Tippe Top's inversion, offering new insights by reformulating the equations and relating them to simpler mechanical systems to better understand its oscillatory dynamics.

Contribution

It introduces a comprehensive analysis of the Tippe Top equations in three forms and links them to simpler systems for improved understanding of inversion dynamics.

Findings

Main equation captures oscillatory motion during inversion

Equations relate Tippe Top to simpler mechanical systems

Provides new analytical approaches to inversion dynamics

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 axis $\mathbf{\hat{3}}$ 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.