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Non-inertial torques and the Euler equation

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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.

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Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Notes

  1. See [2], page 7.

  2. For example, if we consider a system of point charges the internal forces among each pair of these charges is a central force given by Coulomb’s law \({\vec {F}}_{ij} = 1/(4\pi \epsilon _0) \times (q_i q_j) {\vec r}_{ij} / r_{ij}^3\), where \(q_i\) and \(q_j\) are the charges, \({\vec r}_{ij}\) is the separation vector between the charges and \(\epsilon _0\) is the permitivity of vacuum.

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Authors and Affiliations

  1. Department of Mathematics and Physics, SUNY Polytechnic Institute, Utica, NY, 13502, USA

    Amir H. Fariborz

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  1. Amir H. Fariborz
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Correspondence to Amir H. Fariborz.

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Fariborz, A.H. Non-inertial torques and the Euler equation. Eur. Phys. J. Plus 137, 1343 (2022). https://doi.org/10.1140/epjp/s13360-022-03558-x

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