A simple example is furnished by the top ([Mechanics], § 22). The cyclic co-ordinates being ψ, φ, we find
| 2⅋ = Aθ˙², 2K = | (μ − ν cos θ)² | + | ν² | , |
| A sin² θ | C |
2Τ0 = A sin² θψ˙² + C (φ˙ + ψ cos θ)²,
(29)
whence we may verify that ∂Τ0 / ∂θ = −∂K / ∂θ in accordance with (27). And the condition of equilibrium
| ∂K | = − | ∂V |
| ∂θ | ∂θ |
(30)
gives the condition of steady precession.
6. Stability of Steady Motion.
The small oscillations of a conservative system about a configuration of equilibrium, and the criterion of stability, are discussed in [Mechanics], § 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 Über 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.