Non-inertial torques and the Euler equation

Abstract

The key equation describing the rotational dynamics of a rigid body is \({\vec \tau } = \text{d}{\vec L} / dt\) which can be understood based on the Newton’s second and third laws of motion together with the assumption of mutual centrality of the internal forces and is valid in an inertial coordinate system. While this equation is written down by an inertial observer, for practical purposes, it is efficiently worked out within a non-inertial rotating ancillary coordinate system along the principle axes of the rigid body. This results in the famous Euler equation for rotation of the rigid bodies. We show that it is also possible to describe the rotational dynamics of a rigid body from the point of view of a non-inertial observer (rotating with the ancillary coordinate system), provided that the non-inertial torques are taken into account. We explicitly calculate the non-inertial torques and express them in terms of physical characteristics of the rigid body. We show that the resulting dynamical equations exactly recover the Euler equation.