An alternative proof for Euler rotation theorem

TL;DR

This paper presents a new geometric proof of Euler's rotation theorem, demonstrating that any rigid body reconfiguration with a fixed point can be achieved through two successive perpendicular rotations, enhancing understanding of rigid body dynamics.

Contribution

The paper introduces a novel geometric proof of Euler's rotation theorem using successive rotations about perpendicular axes, offering an alternative to existing proofs.

Findings

Provides a new geometric perspective on Euler's rotation theorem

Shows that two successive perpendicular rotations can reconfigure a rigid body

Enhances understanding of rigid body motion and rotation axes

Abstract

Euler's rotation theorem states that any reconfiguration of a rigid body with one of its points fixed is equivalent to a single rotation about an axis passing through the fixed point. The theorem forms the basis for Chasles' theorem which states that it is always possible to represent the general displacement of a rigid body by a translation and a rotation about an axis. Though there are many ways to achieve this, the direction of the rotation axis and angle of rotation are independent of the translation vector. The theorem is important in the study of rigid body dynamics. There are various proofs available for these theorems, both geometric and algebraic. A novel geometric proof of Euler rotation theorem is presented here which makes use of two successive rotations about two mutually perpendicular axis to go from one configuration of the rigid body to the other with one of its points fixed.