Euler's Equation via Lagrangian Dynamics with Generalized Coordinates

TL;DR

This paper derives Euler's equation for rigid body rotation using Lagrangian dynamics and generalized coordinates, including Euler angles and quaternions, providing a unified theoretical framework.

Contribution

It introduces a Lagrangian dynamics approach to derive Euler's equation in generalized coordinates, bridging a gap in the literature.

Findings

Derivation of Euler's equation using Euler angles

Derivation of Euler's equation using quaternions

Unified framework for rigid body dynamics

Abstract

Euler's equation relates the change in angular momentum of a rigid body to the applied torque. This paper fills a gap in the literature by using Lagrangian dynamics to derive Euler's equation in terms of generalized coordinates. This is done by parameterizing the angular velocity vector in terms of 3-2-1 and 3-1-3 Euler angles as well as Euler parameters, that is, unit quaternions.