Analytical solution of the Euler-Poinsot problem

TL;DR

This paper derives an analytical solution for the torque-free motion of a rigid body, explores Poinsot's geometric interpretation, and develops algorithms and animations to visualize these solutions and herpolhodes.

Contribution

It provides a comprehensive analytical and geometric analysis of the Euler-Poinsot problem, including new algorithms for visualization and conditions for closed herpolhodes.

Findings

Analytical expressions for angular velocity and Euler angles.

Animation of Poinsot's geometric solution.

Algorithm for generating closed herpolhodes.

Abstract

In the present paper, an analysis was performed on the torque-free motion of a rigid body, developing Euler's analytical solution and Poinsot's geometric solution. From mathematical formulations, the analytical solution for the time evolution of the angular velocity and Euler's angles was obtained and described given some initial conditions. Besides, an animation of Poinsot's geometric solution was elaborated and a study was carried out on the conditions in which the herpolhode forms a closed curve. Finally, an algorithm was developed in the software Scilab that displays the analytical and numerical solutions obtained, it also generates an animation of the geometric solution, moreover to having an algorithm that generates closed herpolhodes.