Rattleback: a model of how geometric singularity induces dynamic chirality

Abstract

The rattleback is a boat-shaped top with an asymmetric preference in spin. Its dynamics can be described by nonlinearly coupled pitching, rolling, and spinning modes. The chirality, designed into the body as a skewed mass distribution, manifests itself in the quicker transition of $+$spin $\rightarrow$ pitch $\rightarrow$ $-$spin than that of $-$spin $\rightarrow$ roll $\rightarrow$ $+$spin. The curious guiding idea of this work is that we can formulate the dynamics as if a symmetric body were moving in a chiral space. By elucidating the duality of matter and space in the Hamiltonian formalism, we attribute asymmetry to space. The rattleback is shown to live in the space dictated by the Bianchi type ${\rm VI}_{h < -1}$ (belonging to class B) algebra; this particular algebra is used here for the first time in a mechanical example. The class B algebra has a singularity that separates the space (Poisson manifold) into mirror-asymmetric subspaces, breaking the time-reversal symmetry of nearby orbits.