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, ... qm all vanish (see § 5). This has been discussed by Routh, Lord Kelvin and Tait, and Poincaré. 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 + K be a minimum as regards variations of q1, q2, ... qm. 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,
| V + K = Mgh cos θ + | (μ − νcos θ)² | + | ν² | . |
| 2A sin² θ | 2C |
(1)
To examine whether the steady motion with the centre of gravity vertically above the pivot is stable, we must put μ = ν. We then find without difficulty that V + K is a minimum provided ν² ≥ 4AMgh. The method of small oscillations gave us the condition ν² > 4AMgh, and indicated instability in the cases ν² ≤ 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 + K 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 + K 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 Poincaré, 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, ... qm vary. Since the total energy of the system at any instant is given (in the notation of § 5) by an expression of the form ⅋ + V + K, where ⅋ cannot be negative, the argument of Thomson and Tait, given under [Mechanics], § 23, for the statical question, shows that it is a necessary as well as a sufficient condition for secular stability that V + K 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 § 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 − Τ0 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 (ω) about the vertical diameter. This position obviously possesses “ordinary” stability. If a be the radius of the bowl, and θ denote angular distance from the lowest point, we have
V − Τ0 = mga(1 − cos θ) − ½mω²a² sin² θ;
(2)