The small oscillations of a conservative system about a configuration
of equilibrium, and the criterion of stability, are discussed in
MECHANICS, Sec. 23. The question of the stability of given types of
motion is more difficult, owing to the want of a sufficiently general,
and at the same time precise, definition of what we mean by
"stability." A number of definitions which have been propounded by
different writers are examined by F. Klein and A. Sommerfeld in their
work _Ueber die Theorie des Kreisels_ (1897-1903). Rejecting previous
definitions, they base their criterion of stability on the character
of the changes produced in the _path_ of the system by small arbitrary
disturbing impulses. If the undisturbed path be the _limiting form_ of
the disturbed path when the impulses are indefinitely diminished, it
is said to be stable, but not otherwise. For instance, the vertical
fall of a particle under gravity is reckoned as stable, although for a
_given_ impulsive disturbance, however small, the deviation of the
particle's position at any time t from the position which it would
have occupied in the original motion increases indefinitely with t.
Even this criterion, as the writers quoted themselves recognize, is
not free from ambiguity unless the phrase "limiting form," as applied
to a path, be strictly defined. It appears, moreover, that a
definition which is analytically precise may not in all cases be easy
to reconcile with geometrical prepossessions. Thus a particle moving
in a circle about a centre of force varying inversely as the cube of
the distance will if slightly disturbed either fall into the centre,
or recede to infinity, after describing in either case a spiral with
an infinite number of convolutions. Each of these spirals has,
analytically, the circle as its limiting form, although the motion in
the circle is most naturally described as unstable.
A special form of the problem, of great interest, presents itself in
the steady motion of a gyrostatic system, when the non-eliminated
co-ordinates q1, q2, ... q_m all vanish (see Sec. 5). This has been
discussed by Routh, Lord Kelvin and Tait, and Poincare. These writers
treat the question, by an extension of Lagrange's method, as a problem
of small oscillations. Whether we adopt the notion of stability which
this implies, or take up the position of Klein and Sommerfeld, there
is no difficulty in showing that stability is ensured if V + [Kappa]
be a minimum as regards variations of q1, q2, ... q_m. The proof is
the same as that of Dirichlet for the case of statical stability.
We can illustrate this condition from the case of the top, where, in
our previous notation,
( mu - [nu]cos [theta]) squared [nu] squared
V + [Kappa] = Mgh cos[theta] + ---------------------- + -----. (1)
2A sin squared [theta] 2C
To examine whether the steady motion with the centre of gravity
vertically above the pivot is stable, we must put mu = [nu]. We then
find without difficulty that V + [Kappa] is a minimum provided [nu] squared
[>=] 4AMgh. The method of small oscillations gave us the condition
[nu] squared > 4AMgh, and indicated instability in the cases [nu] squared [=<]
4AMgh. The present criterion can also be applied to show that the
steady precessional motions in which the axis has a constant
inclination to the vertical are stable.
The question remains, as before, whether it is _essential_ for
stability that V + [Kappa] should be a minimum. It appears that from
the point of view of the theory of small oscillations it is not
essential, and that there may even be stability when V + [Kappa] is a
maximum. The precise conditions, which are of a somewhat elaborate
character, have been formulated by Routh. An important distinction
has, however, been established by Thomson and Tait, and by Poincare,
between what we may call _ordinary_ or _temporary_ stability (which is
stability in the above sense) and _permanent_ or _secular_ stability,
which means stability when regard is had to possible dissipative
forces called into play whenever the co-ordinates q1, q2, ... q_m
vary. Since the total energy of the system at any instant is given (in
the notation of Sec. 5) by an expression of the form [@] + V +
[Kappa], where [@] cannot be negative, the argument of Thomson
and Tait, given under MECHANICS, Sec. 23, for the statical question,
shows that it is a necessary as well as a sufficient condition for
secular stability that V + [Kappa] should be a minimum. When a system
is "ordinarily" stable, but "secularly" unstable, the operation of the
frictional forces is to induce a gradual increase in the amplitude of
the free vibrations which are called into play by accidental
disturbances.
There is a similar theory in relation to the constrained systems
considered in Sec. 3 above. The equation (21) there given leads to the
conclusion that for secular stability of any type of motion in which
the velocities q`1, q`2, ... q`_n are zero it is necessary and
sufficient that the function V - [Tau]0 should be a minimum.
The simplest possible example of this is the case of a particle at the
lowest point of a smooth spherical bowl which rotates with constant
angular velocity ([omega]) about the vertical diameter. This position
obviously possesses "ordinary" stability. If a be the radius of the
bowl, and [theta] denote angular distance from the lowest point, we
have
V - [Tau]0 = mga(1 - cos [theta]) - 1/2m[omega] squareda squared sin squared [theta]; (2)
this is a minimum for [theta] = 0 only so long as [omega] squared < g/a. For
greater values of [omega] the only position of "permanent" stability
is that in which the particle rotates with the bowl at an angular
distance cos^(-1) (g/[omega] squareda) from the lowest point. To examine the
motion in the neighbourhood of the lowest point, when frictional
forces are taken into account, we may take fixed ones, in a horizontal
plane, through the lowest point. Assuming that the friction varies as
the relative velocity, we have
x" = -p squaredx - k(x` + [omega]y), \ (3)
y" = -p squaredy - k(y` - [omega]x), /
where p squared = g/a. These combine into
z" + kz` + (p squared - ik[omega])z = 0, (4)
where z = x + iy, i = [root]-1. Assuming z = Ce^([lambda]t), we find
[lambda] = -1/2k(1 [-+] [omega]/p) +- ip, (5)
if the square of k be neglected. The complete solution is then
x + iy = C1e^(-ss1t)e^(ipt) + C2e^(-ss2t)e^(-ipt), (6)
where ss1 = 1/2k(1 - [omega]/p), ss2 = 1/2k(1 + [omega]/p). (7)
This represents two superposed circular vibrations, in opposite
directions, of period 2[pi]/p. If [omega] < p, the amplitude of each
of these diminishes asymptotically to zero, and the position x = 0, y
= 0 is permanently stable. But if [omega] > p the amplitude of that
circular vibration which agrees in sense with the rotation [omega]
will continually increase, and the particle will work its way in an
ever-widening spiral path towards the eccentric position of secular
stability. If the bowl be not spherical but ellipsoidal, the vertical
diameter being a principal axis, it may easily be shown that the
lowest position is permanently stable only so long as the period of
the rotation is longer than that of the slower of the two normal
modes in the absence of rotation (see MECHANICS, Sec. 13).