x + iy = C1e−β1t eipt + + C2e−β2t e−ipt,
(6)
where
β1 = ½k (1 − ω/p), β2 = ½k (1 + ω/p).
(7)
This represents two superposed circular vibrations, in opposite directions, of period 2π/p. If ω < p, the amplitude of each of these diminishes asymptotically to zero, and the position x = 0, y = 0 is permanently stable. But if ω > p the amplitude of that circular vibration which agrees in sense with the rotation ω 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], § 13).
7. Principle of Least Action.
The preceding theories give us statements applicable to the system at any one instant of its motion. We now come to a series of theorems relating to the whole motion of the system between any two configurations through which it passes, Stationary Action. viz. we consider the actual motion and compare it with other imaginable motions, differing infinitely little from it, between the same two configurations. We use the symbol δ to denote the transition from the actual to any one of the hypothetical motions.
The best-known theorem of this class is that of Least Action, originated by P.L.M. de Maupertuis, but first put in a definite form by Lagrange. The “action” of a single particle in passing from one position to another is the space-integral of the momentum, or the time-integral of the vis viva. The action of a dynamical system is the sum of the actions of its constituent particles, and is accordingly given by the formula
A = Σ ∫ mvds = Σ ∫ mv²dt = 2 ∫ Τdt.