A Poincar\'e section for the general heavy rigid body

TL;DR

This paper introduces a general method for analyzing rigid body dynamics using Poincaré sections, identifying their topological types and proposing a projection for visualization.

Contribution

It develops a universal recipe for constructing Poincaré sections in rigid body dynamics and characterizes their topological structures.

Findings

Identifies possible topologies of Poincaré surfaces for rigid bodies.

Proposes a projection method onto a torus for visualization.

Provides a systematic approach for studying energy surface trajectories.

Abstract

A general recipe is developed for the study of rigid body dynamics in terms of Poincar\'e surfaces of section. A section condition is chosen which captures every trajectory on a given energy surface. The possible topological types of the corresponding surfaces of section are determined, and their 1:1 projection to a conveniently defined torus is proposed for graphical rendering.