Since θ, ω are the polar co-ordinates (in a horizontal plane) of a point on the axis of symmetry, relative to an initial line which revolves with constant angular velocity ν/2A, we see by comparison with § 14 (15) (16) that the motion of such a point will be elliptic-harmonic superposed on a uniform rotation ν/2A, provided ν2 > 4AMgh. This gives (in essentials) the theory of the “gyroscopic pendulum.”
§ 23. Stability of Equilibrium. Theory of Vibrations.—If, in a conservative system, the configuration (q1, q2, ... qn) be one of equilibrium, the equations (14) of § 22 must be satisfied by q̇1, q̇2 ... q̇n = 0, whence
∂V / ∂qr = 0.
(1)
A necessary and sufficient condition of equilibrium is therefore that the value of the potential energy should be stationary for infinitesimal variations of the co-ordinates. If, further, V be a minimum, the equilibrium is necessarily stable, as was shown by P. G. L. Dirichlet (1846). In the motion consequent on any slight disturbance the total energy T + V is constant, and since T is essentially positive it follows that V can never exceed its equilibrium value by more than a slight amount, depending on the energy of the disturbance. This implies, on the present hypothesis, that there is an upper limit to the deviation of each co-ordinate from its equilibrium value; moreover, this limit diminishes indefinitely with the energy of the original disturbance. No such simple proof is available to show without qualification that the above condition is necessary. If, however, we recognize the existence of dissipative forces called into play by any motion whatever of the system, the conclusion can be drawn as follows. However slight these forces may be, the total energy T + V must continually diminish so long as the velocities q̇1, q̇2, ... q̇n differ from zero. Hence if the system be started from rest in a configuration for which V is less than in the equilibrium configuration considered, this quantity must still further decrease (since T cannot be negative), and it is evident that either the system will finally come to rest in some other equilibrium configuration, or V will in the long run diminish indefinitely. This argument is due to Lord Kelvin and P. G. Tait (1879).
In discussing the small oscillations of a system about a configuration of stable equilibrium it is convenient so to choose the generalized cc-ordinates q1, q2, ... qn that they shall vanish in the configuration in question. The potential energy is then given with sufficient approximation by an expression of the form
2V = c11q12 + c22q22 + ... + 2c12q1q2 + ...,
(2)
a constant term being irrelevant, and the terms of the first order being absent since the equilibrium value of V is stationary. The coefficients crr, crs are called coefficients of stability. We may further treat the coefficients of inertia arr, ars of § 22 (1) as constants. The Lagrangian equations of motion are then of the type
a1rq̈1 + a2rq̈2 + ... + anrq̈n + c1rq1 + c2rq2 + ... + cnrqn = Qr,