On the rattleback dynamics

TL;DR

This paper explores the complex dynamical behavior of an idealized conservative rattleback model using Hamilton-Poisson structures, offering geometric insights and a method for orbit stabilization.

Contribution

It provides a geometric characterization of the rattleback's orbit space and introduces an explicit stabilization method for arbitrary orbits.

Findings

Geometric characterization of orbit space via Whitney stratifications

Explicit method for asymptotic stabilization of orbits

Analysis of dynamical properties using Hamilton-Poisson framework

Abstract

In this paper we present some relevant dynamical properties of an idealized conservative model of the rattleback, from the Poisson dynamics point of view. In the first half of the article, along with a dynamical study of the orbits, using a Hamilton-Poisson realization of the dynamical system, we provide a geometric characterization of the space of orbits in terms of Whitney stratifications associated to the image of the energy-Casimir mapping. In the second half of the article we provide an explicit method to stabilize asymptotically any arbitrary fixed orbit/cycle of the rattleback system and to keep unchanged the geometry of the model space.