XLI Workshop on Geometric Methods in Physics Białystok, 1-6.07.2024 XIII School on Geometry and Physics Białystok, 8-12.07.2024

# Stefan Rauch

## Understanding reversals of a Rattleback

The rattleback is a rigid body having a boat like shape (modelled as a bottom half of an 3-axial ellipsoid) having asymetric (chiral) distribution of mass. When the rattleback is spun on its bottom in the “wrong” direction then it starts to rattle, it slows down and acquires rotation in the opposite, preferred sense of spinning.
This behaviour defies our intuition about conservation of angular momentum as the force and the torque responsible for changing the angular momentum (and the direction of spinning) is not easily discernible.
Overwhelming majority of papers on the rattleback motion study the dependence of stability for spinning solutions: on the sense of rotation, on the shape of rattleback´s surface and on the distribution of mass. There has been no simple explanation of the rattleback behaviour in terms of physical forces and torques.

This question has been the subject of our paper with M.Przybylska that have appeared in Regular and Chaotic Dynamics, a journal of Russian Academy of Sciences. In this paper we study the motion of a toy rattleback by using frictionless Newton equations of motion for a rigid body rolling without sliding in a plane. In these equations it is the reaction force of the supporting surface that is the source of the torque turning the rattleback in the preferred sense of rotation.
The picture is, however, more subtle as it appears that the direction of the torque depends on the initial conditions and a frictionless, low energy rattleback admits reversals in both directions (!).

I will present a simple, intuitive understanding of how the rattleback´s motion depends on initial conditions and will discuss how it is consistent with numerical simulations of rattlebacks equations for tapping and for spinning initial conditions.
Simulations show also that long time behaviour of such rattleback is, for low energy initial conditions, quasi-periodic and there are inifinitly many reversals in both directions.