Or, taking Example (B) of § 319, and putting mP for P, and. mh for h, Hence, putting u= a, and making h2 = gm'/ma3 so that motion -1 may be possible, we find Hence the circular motion is always stable; and the period of surface. 351. The case of a particle moving on a smooth fixed surface Kinetic stnbility of a under the influence of no other force than that of the con- particle moving on straint, and therefore always moving along a geodetic line of a smooth the surface, affords extremely simple illustrations of stability and instability. For instance, a particle placed on the inner circle of the surface of an anchor-ring, and projected in the plane of the ring, would move perpetually in that circle, but unstably, as the smallest disturbance would clearly send it away from this path, never to return until after a digression round the outer edge. (We suppose of course that the particle is held to the surface, as if it were placed in the infinitely narrow space between a solid ring and a hollow one enclosing it.) But if a particle is placed on the outermost, or greatest, Kinetic stability of moving on a smooth surface. π circle of the ring, and projected in its plane, an infinitely small a particle disturbance will cause it to describe a sinuous path cutting the circle at points round it successively distant by angles each equal toba, or intervals of time, √b/w Ja, where a denotes the radius of that circle, w the angular velocity in it, and b the radius of the circular cross section of the ring. This is proved by remarking that an infinitely narrow band from the outermost part of the ring has, at each point, a and b for its principal radii of curvature, and therefore (§ 150) has for its geodetic lines the great circles of a sphere of radius ab, upon which (§ 152) it may be bent. Kinetic sta bility. In rable oscil lations. 352. In all these cases the undisturbed motion has been circular or rectilineal, and, when the motion has been stable, the effect of a disturbance has been periodic, or recurring with the same phases in equal successive intervals of time. An illustration of thoroughly stable motion in which the effect of a disturbance is not " periodic," is presented by a particle sliding down an inclined groove under the action of gravity. To take the simplest case, we may consider a particle sliding down along the lowest straight line of an inclined hollow cylinder. If slightly disturbed from this straight line, it will oscillate on each side of it perpetually in its descent, but not with a uniform periodic motion, though the durations of its excursions to each side of the straight line are all equal. 353. A very curious case of stable motion is presented by commensu a particle constrained to remain on the surface of an anchorring fixed in a vertical plane, and projected along the great circle from any point of it, with any velocity. An infinitely small disturbance will give rise to a disturbed motion of which the path will cut the vertical circle over and over again for ever, at unequal intervals of time, and unequal angles of the circle; and obviously not recurring periodically in any cycle, except with definite particular values for the whole energy, some of which are less and an infinite number are greater than that which just suffices to bring the particle to the highest point of the ring. The full mathematical investigation of these circumstances would afford an excellent exercise in the theory of differential equations, but it is not necessary for our present illustrations. kinetic sta 354. In this case, as in all of stable motion with only two Oscillatory degrees of freedom, which we have just considered, there has bility. been stability throughout the motion; and an infinitely small disturbance from any point of the motion has given a disturbed path which intersects the undisturbed path over and over again at finite intervals of time. But, for the sake of simplicity at present confining our attention to two degrees of freedom, we have a limited stability in the motion of an unresisted pro- Limited jectile, which satisfies the criterion of stability only at points bility. of its upward, not of its downward, path. Thus if MOPQ be • M stability of the path of a projectile, and if at 0 it be disturbed by an infi- Kinetic nitely small force either way perpendicular to its instantaneous a projectile. direction of motion, the disturbed path will cut the undisturbed infinitely near the point P where the direction of motion is perpendicular to that at 0: as we easily see by considering that the line joining two particles projected from one point at the same instant with equal velocities in the directions of any two lines, will always remain perpendicular to the line bisecting the angle between these two lines. General criterion. 355. The principle of varying action gives a mathematical criterion for stability or instability in every case of motion. Thus in the first place it is obvious, and it will be proved below (S$ 358, 361), that if the action is a true minimum in the motion. of a system from any one configuration to the configuration reached at any other time, however much later, the motion is Examples. thoroughly unstable. For instance, in the motion of a particle constrained to remain on a smooth fixed surface, and uninfluenced by gravity, the action is simply the length of the path, multiplied by the constant velocity. Hence in the particular case of a particle uninfluenced by gravity, moving round the inner circle in the plane of an anchor-ring considered above, the action, or length of path, is clearly a minimum from any one point to the point reached at any subsequent time. (The action is not merely a minimum, but is the smaller of two minimums, when the course is from any point of the circular path to any other, through less than half a circumference of the circle.) On the other hand, although the path from any point in the greatest circle of the ring to any other at a distance from it along the circle, less than π√ab, is clearly least possible if along the circumference; the path of absolutely least length is not along the circumference between two points at a greater circular distance than Tab from one another, nor is the path along the circumference between them a minimum at all in this latter case. On any surface whatever which is everywhere anticlastic, or along a geodetic of any surface which passes altogether through an anticlastic region, the motion is thoroughly unstable. For if it were stable from any point 0, we should have the given undisturbed path, and the disturbed path from 0 cutting it at some point Q;-two different geodetic lines joinMotion of a ing two points; which is impossible on an anticlastic surface, an anticlas- inasmuch as the sum of the exterior angles of any closed unstable; figure of geodetic lines exceeds four right angles (§ 136) when the integral curvature of the enclosed area is negative, which (S$ 138, 128) is the case for every portion of surface thoroughly anticlastic. But, on the other hand, it is easily proved that if we have an endless rigid band of curved surface everywhere synclastic, with a geodetic line running through its Motion on an anticlastic surface proved unstable. particle on tic surface, clastic sur middle, the motion of a particle projected along this line will on a synbe stable throughout, and an infinitely slight disturbance will face, stable. give a disturbed path cutting the given undisturbed path again and again for ever at successive distances differing according to the different specific curvatures of the intermediate portions of the surface. If from any point, N, of the undisturbed path, a perpendicular be drawn to cut the infinitely near disturbed path in E, the angles OEN and NOE N must (138) be toge equation of ther greater than a right angle by an amount equal to the in- Differential tegral curvature of the area EON. From this the differential disturbed equation of the disturbed path may be obtained immediately. Let EON=a, ON=s, and NE=u; and let 9, a known function of 8, be the specific curvature (§ 136) of the surface in the neighbourhood of N. Let also, for a moment, & denote the complement of the angle OEN. We have a-= Duds. path. Hence But, obviously, hence L'ouds. When is constant (as in the case of the equator of a surface of u = A cos (8 √≈ + E), agreeing with the result (§ 351) which we obtained by develop- The case of two or more bodies supported on parallel axes in the manner explained above in § 348, and rotating with the centre of inertia of the whole at the least possible distance from the fixed axis, affords a very good illustration also of this proposition which may be safely left as an exercise to the student. |