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Conservation laws and Liapounov stability of the free rotation of a rigid body

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Abstract

The paper derives the well known stabilities of free rotation of a rigid body about its principal axes of least and greatest moments of inertia directly from the constancy of the kinetic energy and of the square of the angular momentum. The resulting proof of Liapounov stability yields new quantitative measures of this stability. Involving only simple algebra, it depends on satisfying a weak sufficient condition that insures an unchanging sign of the main component of the angular velocity ω. The method cannot be used, however, to prove the well known instability of rotation about the intermediate axis.

The quantitative results for the radii of the spheres in ω-space that occur in the Liapounov proof lead to a physical result that may be of interest. If the earth were truly a rigid body, rotating freely, the angular deviation of its instantaneous polar axis from the nearest principal axis could not increase from a given initial value by more than the factor ℚ2.

These same quantitative results for the radii of the Liapounov spheres in ω-space also lead to suflicient conditions for the rotational stability of almost spherical bodies of various shapes, prolate or oblate. They may be pertinent in designing ‘spheres’ to be used in currently planned experiments to test general relativity by observing the rate of precession of a rotating sphere in orbit about the earth.

The above results follow from restricted Liapounov stability alone. The last section contains the proof of general Liapounov stability.

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Authors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, Mass., USA

    John P. Vinti

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  1. John P. Vinti
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This paper was prepared under the sponsorship of the Electronics Research Center of the National Aeronautics and Space Administration through NASA Grant NGR 22-009-262.

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Vinti, J.P. Conservation laws and Liapounov stability of the free rotation of a rigid body. Celestial Mechanics 1, 59–71 (1969). https://doi.org/10.1007/BF01230633

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  • DOI: https://doi.org/10.1007/BF01230633

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