On a Dynamical Top,
for exhibiting the phenomena of the motion of a system of invariable
form about a fixed point, with some suggestions as to the Earth’s
motion
James Clerk Maxwell
[From the _Transactions of the Royal Society of Edinburgh_, Vol. XXI.
Part IV.]
(Read 20th April, 1857.)
To those who study the progress of exact science, the common
spinning-top is a symbol of the labours and the perplexities of men who
had successfully threaded the mazes of the planetary motions. The
mathematicians of the last age, searching through nature for problems
worthy of their analysis, found in this toy of their youth, ample
occupation for their highest mathematical powers.
No illustration of astronomical precession can be devised more perfect
than that presented by a properly balanced top, but yet the motion of
rotation has intricacies far exceeding those of the theory of
precession.
Accordingly, we find Euler and D’Alembert devoting their talent and
their patience to the establishment of the laws of the rotation of
solid bodies. Lagrange has incorporated his own analysis of the problem
with his general treatment of mechanics, and since his time M. Poinsôt
has brought the subject under the power of a more searching analysis
than that of the calculus, in which ideas take the place of symbols,
and intelligible propositions supersede equations.
In the practical department of the subject, we must notice the rotatory
machine of Bohnenberger, and the nautical top of Troughton. In the
first of these instruments we have the model of the Gyroscope, by which
Foucault has been able to render visible the effects of the earth’s
rotation. The beautiful experiments by which Mr J. Elliot has made the
ideas of precession so familiar to us are performed with a top, similar
in some respects to Troughton’s, though not borrowed from his.
The top which I have the honour to spin before the Society, differs
from that of Mr Elliot in having more adjustments, and in being
designed to exhibit far more complicated phenomena.
The arrangement of these adjustments, so as to produce the desired
effects, depends on the mathematical theory of rotation. The method of
exhibiting the motion of the axis of rotation, by means of a coloured
disc, is essential to the success of these adjustments. This optical
contrivance for rendering visible the nature of the rapid motion of the
top, and the practical methods of applying the theory of rotation to
such an instrument as the one before us, are the grounds on which I
bring my instrument and experiments before the Society as my own.
I propose, therefore, in the first place, to give a brief outline of
such parts of the theory of rotation as are necessary for the
explanation of the phenomena of the top.
I shall then describe the instrument with its adjustments, and the
effect of each, the mode of observing of the coloured disc when the top
is in motion, and the use of the top in illustrating the mathematical
theory, with the method of making the different experiments.
Lastly, I shall attempt to explain the nature of a possible variation
in the earth’s axis due to its figure. This variation, if it exists,
must cause a periodic inequality in the latitude of every place on the
earth’s surface, going through its period in about eleven months. The
amount of variation must be very small, but its character gives it
importance, and the necessary observations are already made, and only
require reduction.
On the Theory of Rotation.
The theory of the rotation of a rigid system is strictly deduced from
the elementary laws of motion, but the complexity of the motion of the
particles of a body freely rotating renders the subject so intricate,
that it has never been thoroughly understood by any but the most expert
mathematicians. Many who have mastered the lunar theory have come to
erroneous conclusions on this subject; and even Newton has chosen to
deduce the disturbance of the earth’s axis from his theory of the
motion of the nodes of a free orbit, rather than attack the problem of
the rotation of a solid body.
The method by which M. Poinsôt has rendered the theory more manageable,
is by the liberal introduction of “appropriate ideas,” chiefly of a
geometrical character, most of which had been rendered familiar to
mathematicians by the writings of Monge, but which then first became
illustrations of this branch of dynamics. If any further progress is to
be made in simplifying and arranging the theory, it must be by the
method which Poinsôt has repeatedly pointed out as the only one which
can lead to a true knowledge of the subject,--that of proceeding from
one distinct idea to another instead of trusting to symbols and
equations.
An important contribution to our stock of appropriate ideas and methods
has lately been made by Mr R. B. Hayward, in a paper, “On a Direct
Method of estimating Velocities, Accelerations, and all similar
quantities, with respect to axes, moveable in any manner in Space.”
(_Trans. Cambridge Phil. Soc_ Vol. x. Part I.)
* In this communication I intend to confine myself to that part of the
subject which the top is intended io illustrate, namely, the alteration
of the position of the axis in a body rotating freely about its centre
of gravity. I shall, therefore, deduce the theory as briefly as
possible, from two considerations only,--the permanence of the original
_angular momentum_ in direction and magnitude, and the permanence of
the original _vis viva_.
* The mathematical difficulties of the theory of rotation arise chiefly
from the want of geometrical illustrations and sensible images, by
which we might fix the results of analysis in our minds.
It is easy to understand the motion of a body revolving about a fixed
axle. Every point in the body describes a circle about the axis, and
returns to its original position after each complete revolution. But if
the axle itself be in motion, the paths of the different points of the
body will no longer be circular or re-entrant. Even the velocity of
rotation about the axis requires a careful definition, and the
proposition that, in all motion about a fixed point, there is always
one line of particles forming an instantaneous axis, is usually given
in the form of a very repulsive mass of calculation. Most of these
difficulties may be got rid of by devoting a little attention to the
mechanics and geometry of the problem before entering on the discussion
of the equations.
Mr Hayward, in his paper already referred to, has made great use of the
mechanical conception of Angular Momentum.
Definition 1 The Angular Momentum of a particle about an axis is
measured by the product of the mass of the particle, its velocity
resolved in the normal plane, and the perpendicular from the axis on
the direction of motion.
* The angular momentum of any system about an axis is the algebraical
sum of the angular momenta of its parts.
As the _rate of change_ of the _linear momentum_ of a particle measures
the _moving force_ which acts on it, so the _rate of change_ of
_angular momentum_ measures the _moment_ of that force about an axis.
All actions between the parts of a system, being pairs of equal and
opposite forces, produce equal and opposite changes in the angular
momentum of those parts. Hence the whole angular momentum of the system
is not affected by these actions and re-actions.
* When a system of invariable form revolves about an axis, the angular
velocity of every part is the same, and the angular momentum about the
axis is the product of the _angular velocity_ and the _moment of
inertia_ about that axis.
* It is only in particular cases, however, that the _whole_ angular
momentum can be estimated in this way. In general, the axis of angular
momentum differs from the axis of rotation, so that there will be a
residual angular momentum about an axis perpendicular to that of
rotation, unless that axis has one of three positions, called the
principal axes of the body.
By referring everything to these three axes, the theory is greatly
simplified. The moment of inertia about one of these axes is greater
than that about any other axis through the same point, and that about
one of the others is a minimum. These two are at right angles, and the
third axis is perpendicular to their plane, and is called the mean
axis.
* Let $A$, $B$, $C$ be the moments of inertia about the principal axes
through the centre of gravity, taken in order of magnitude, and let
$\omega_1$ $\omega_2$ $\omega_3$ be the angular velocities about them,
then the angular momenta will be $A\omega_1$, $B\omega_2$, and
$C\omega_3$.
Angular momenta may be compounded like forces or velocities, by the law
of the “parallelogram,” and since these three are at right angles to
each other, their resultant is
\begin{displaymath} \sqrt{A^2\omega_1^2 + B^2\omega_2^2 +
C^2\omega_3^2} = H \end{displaymath} (1)
and this must be constant, both in magnitude and direction in space,
since no external forces act on the body.
We shall call this axis of angular momentum the _invariable axis_. It
is perpendicular to what has been called the invariable plane. Poinsôt
calls it the axis of the couple of impulsion. The _direction-cosines_
of this axis in the body are,
\begin{displaymath} \begin{array}{c c c} \displaystyle l =
\frac{A\omega_1}{H}, ... ...ga_2}{H}, & \displaystyle n =
\frac{C\omega_3}{H}. \end{array}\end{displaymath}
Since $I$, $m$ and $n$ vary during the motion, we need some additional
condition to determine the relation between them. We find this in the
property of the _vis viva_ of a system of invariable form in which
there is no friction. The _vis viva_ of such a system must be constant.
We express this in the equation
\begin{displaymath} A\omega_1^2 + B\omega_2^2 + C\omega_3^2 = V
\end{displaymath} (2)
Substituting the values of $\omega_1$, $\omega_2$, $\omega_3$ in terms
of $l$, $m$, $n$,
\begin{displaymath} \frac{l^2}{A} + \frac{m^2}{B} + \frac{n^2}{C} =
\frac{V}{H^2}. \end{displaymath}
Let $1/A = a^2$, $1/B = b^2$, $1/c = c^2$, $V/H^2 = e^2$, and this
equation becomes
\begin{displaymath} a^2l^2 + b^2m^2 + c^2n^2 = e^2
\end{displaymath} (3)
and the equation to the cone, described by the invariable axis within
the body, is
\begin{displaymath} (a^2 - e^2) x^2 + (b^2 - e^2) y^2 + (c^2 - e^2) z^2
= 0 \end{displaymath} (4)
The intersections of this cone with planes perpendicular to the
principal axes are found by putting $x$, $y$, or $z$, constant in this
equation. By giving $e$ various values, all the different paths of the
pole of the invariable axis, corresponding to different initial
circumstances, may be traced.
Figure: Figure 1
* In the figures, I have supposed $a^2 = 100$, $b^2= 107$, and $c^2=
110$. The first figure represents a section of the various cones by a
plane perpendicular to the axis of $x$, which is that of greatest
moment of inertia. These sections are ellipses having their major axis
parallel to the axis of $b$. The value of $e^2$ corresponding to each
of these curves is indicated by figures beside the curve. The
ellipticity increases with the size of the ellipse, so that the section
corresponding to $e^2 = 107$ would be two parallel straight lines
(beyond the bounds of the figure), after which the sections would be
hyperbolas.