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VIII.—On the Rotation of a Rigid Body about a Fixed Point

Published online by Cambridge University Press:  17 January 2013

Tait
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Extract

Although it is very improbable that there remains to be discovered any new, and at the same time simple, fact connected with a question which has been elaborately treated by many of the greatest mathematicians of this and the preceding century, the employment of a new mathematical method may enable us to present some of their results in a more intelligible form, and with far less expenditure of analytical power than has hitherto been deemed necessary; and it may give us such an insight into the question, that we shall be able easily to discover the mutual relations among the various processes which have been already employed; so far, at least, as these differ in principle, and not merely in the peculiar co-ordinates assumed for the purpose of simplifying the equations. Such a method is that of Quaternions, which seems to be expressly fitted for the symmetrical evolution of truths which are usually obtained by the ordinary Cartesian methods only after great labour of calculation, and by modes of attack so indirect, and at first sight so purposeless, as to bewilder all but a very small class of readers. Quaternions afford so clear a view of the nature of the question they are applied to, that even the student, if he have some little knowledge of them, can often see why a transformation is made, whose object he would have been unable to discover had the problem been masked in the unnecessarily artificial difficulties of Cartesian geometry, or the outrageously repulsive formulæ of spherical trigonometry.

Type
Research Article
Information
Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 25 , Issue 2 , 1869 , pp. 261 - 303
Copyright
Copyright © Royal Society of Edinburgh 1869

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References

page 263 note * Report on the Progress of the Solution of certain Special Problems of Dynamics.—Brit. Ass. Report, 1862.Google Scholar

page 263 note † Proc. R. I. A., 1846. See also §§ 1 and 4 below.

page 263 note ‡ See also Cambridge and Dublin Math. Journal, vol i. (1846)Google Scholar.

page 265 note * Proc. R. I. A. November 11, 1844.

page 265 note † Phil. Mag. Feb. 1845.

page 266 note * Phil. Mag., Sept. 1848.

page 277 note * To these it is unnecessary to add

Tq = constant,

as this constancy of Tq is proved by the form of (7). For, had Tq been variable, there must have been a quaternion in place of the vector η. In fact, = 2S · qKq = (Tq)2 Sη = 0.

page 279 note * For further information about this equation, see Hamilton, Proc. R. I. A. 1847, and Elements of Quaternions, p. 755. Also Tait, Quaternions, § 367.

page 280 note * [Inserted Dec. 19, 1868.] I have lately found that Hamilton, , in his Elements of Quaternions (1866)Google Scholar, has obtained this equation in a manner almost identical with that last given.

page 282 note * To get an idea of the nature of this equation, let us integrate it on the supposition that η is a constant vector. By differentiation and substitution, we get

Hence

Substituting in the given equation we have

Hence

which are virtually the same equation—and thus

And the interpretation of q( )q −1 will obviously then be a rotation about η through the angle tTη, together with any other arbitrary rotation whatever. Thus any position whatever may be taken as the initial one of the body—and Q1 ( ) Q1−1 brings it to its required position at time t = 0.

page 291 note * In fact any equation such as

where ψ is a constant self-conjugate linear and vector function, gives

whence

where ν represents the normal-vector. For its locus, we have

and by substitution for ζ and ψζ in the given equation, we have

page 292 note * See Cayley, B. A. Report, 1862.

page 298 note * The tensor of q has been assumed constant. Accordingly we find by this formula

page 300 note * If m be the mass of the pendulum bob, α the vector representing the string, its tension, and γ′ the acceleration due to gravity

or, eliminating ,

It is well to observe that this is the equation of motion of a pendulum bob, acted on by no forces, if − γ′ be the acceleration of the point of suspension.