192
Non-Spherical Masses—Dynamical Principles [ch. yii
thick lines represent series of stable configurations, and thin lines series of
unstable configurations, the series PQ being assumed to be stable in every case
173. Suppose that in any physical problem /x changes very slowly, the
direction of change being that represented by an upward movement in our
diagrams. From what has already been said, it is clear that the following rule
will trace out the sequence of stable states which will be followed by the
system as /Lt varies.
Start from a configuration in the diagram which is known to be stable
and follow a path along linear series of equilibrium so as always to move
upwards, and so as always to cross over from one series to another at a point
of bifurcation. So long as we can do this we are following a sequence of con
figurations which is always stable. When it becomes impossible to do this any
longer, a value of /x has been reached beyond which no stable configurations
exist, and if the physical conditions continue to change so that /x attains to a
still higher value, the statical problem gives place to a dynamical one; it is
no longer a question of tracing out a sequence of gradual secular changes, but
of following up a comparatively rapid motion of a cataclysmic nature.
At each point of bifurcation there is necessarily a certain amount of
indefiniteness in the path which will actually be followed. For instance, in
fig. 20 (i), the system on arriving at Q may proceed either along QT or along
QR, both being equally consistent with the maintenance of stability, and so
far as can be seen equally likely.
This complication causes no difficulty in actual problems. It arises from
the obvious circumstance that a general discussion of stability, although
competent to determine when stability ceases, cannot in general determine
what will happen after stability has ceased. A general discussion of stability
will readily shew that a top spinning slowly on its point is in unstable equi
librium, but it cannot determine in which precise direction the top will first
fall to the ground.
174. In his classical paper* in which the theory of linear series and points
of bifurcation was first developed, Poincare used analytical methods to obtain
results identical with those just given.
If /x is the only parameter which can vary, the potential energy W of the
system may be written in the form
0«* ••• 0n, p)
and the configurations of equilibrium are given by the equations
••• 0», /0-0, etc. (174-1).
* l.c. ante.