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Euler’s Theorem on Rigid Body Displacements Generalized to n Dimensions

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Abstract

Euler’s Theorem on rigid body displacements in three dimensions was generalized to n dimensions by Schoute late in the 19th century. The generalization states that any displacement of a rigid body about a fixed point in n dimensions can be achieved by floor(n/2) simple rotations in mutually orthogonal planes about the fixed point. Unfortunately, this elegant generalization is difficult to find in modern textbooks. It is treated here for the benefit of the present readership. The theorem is proved using the Schur decomposition of an orthogonal matrix. In addition the canonical form of a skew symmetric matrix is derived and then used to show that the angular velocity of a rigid body in n dimensions is composed of floor(n/2) angular rates in mutually orthogonal planes.

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  1. PMB 272, 13 Summit Square, Rt. 413 & Doublewoods Rd., Langhorne, PA, 19047, USA

    Robert Bauer

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  1. Robert Bauer
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Correspondence to Robert Bauer.

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Bauer, R. Euler’s Theorem on Rigid Body Displacements Generalized to n Dimensions. J of Astronaut Sci 50, 305–309 (2002). https://doi.org/10.1007/BF03546254

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  • DOI: https://doi.org/10.1007/BF03546254

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