710 MECHANICS But suppose to become e0 lt where e is a constant, and we have 0" = <?!, which is Newton s hypothesis. The above equations become or, as they may be written, From these the proposition is obvious. Other examples of central orbits will be given when we discuss general principles, such as " least action " and "varying action." Special Problem. The Brachistochrone. Bracbis- 150. A celebrated problem in the history of dynamics is tockrone. that of the "curve of swiftest descent," as it was called: Two points being given, which are neither in a vertical nor in a horizontal line, to find the curve joining them down ivhich a particle sliding under gravity, and starting from rest at the higher, ivill reach the other in the least possible time. The curve must evidently lie in the vertical plane passing through the points. For suppose it not to lie in that plane, project it orthogonally on the plane, and call corresponding elements of the curve and its projection cr and cr . Then if a particle slide down the projected curve its speed at a- will be the same as the speed in the other at cr. But cr is never less than cr , and is generally greater. Hence the time through a- is generally less than that through cr, and never greater. That is, the whole time of falling through the projected curve is less than that through the curve itself. Or the required curve lies in the vertical plane through the points. Also it is easy to see that, if the time of descent through the entire curve is a minimum, that through any portion of the curve is less than if that portion were changed into any other curve. Condi- And it is obvious that, betiveen any two contiguous equal tions for va i ue$ O f a continuously varying quantity, a maximum or minimum must lie. [This principle, though excessively simple (witness its application to the barometer or thermometer), is of very great power, and often enables us to solve problems of maxima and minima, such as require, in analysis, not merely the processes of the differential calculus but those of the calculus of variations. The present is a good example.] Let, then, PQ, QR and PQ , Q R (fig. 48) be two pairs of indefinitely small sides of polygons such that the time of descending through either pair, starting from P with a given speed, may be equal. Let QQ be hori zontal, and indefinitely small compared with PQ and QR, The brachisto chrone must lie between these paths, and must possess any property which they possess in common. Hence if v be the speed down PQ (supposed uniform) and v that down QE, draw- a maxi mum. 48. ing Qm, Q n perpendicular to RQ , PQ, we must have Now if be the inclination of PQ to the horizon, & that of QR, Qn = QQ cos#, Q ?>i = QQ cos$ . Hence the above equation becomes cos 9 cos 6 v v This is true for any two consecutive elements of the required curve ; and therefore throughout the curve v txcosO. But v 2 cf. vertical distance fallen through ( 28). Hence the curve required is such that the cosine of the angle it makes with the horizontal line through the point of departure varies as the square root of the distance from that line, which is easily seen to be a property of the cycloid, if we remember that the tangent to that curve is parallel to the corresponding chord of its generating circle. For in fig. 45, 137, cos BP M = cos BAP = ~ = A/ Jg VAM . The brachistochrone then, under gravity, is an inverted cycloid whose cusp is at the point from which the particle descends. 151, Whatever be the impressed forces, reasoning similar to that in last section would show that the oscu lating plane of the brachistochrone always contains the resultant force, and that where 6 is now the complement of the angle between the curve and the resultant of the impressed forces. Let that resultant = F, and let the element PQ = 8s, and O = d + 50. Then But v a cos# ; which gives Sv sin0 o cos 6 Hence But in the limit =/>, the radius of absolute curvature at Q; 50 and Fcos0 is the normal component of the impressed force. Hence we obtain the result that, in any brachistochrone, the pressure on the curve is double of that due to the force acting. 152. Now for the unconstrained path from P to R we la&vejvds a minimum ( 202). Hence in the same way as before, <p being the angle corresponding to 6, vcos<j> = v cos({> from element to element, and therefore throughout the curve, if the direction of the force Le constant. Now, if the velocities in the unconstrained and brachis tochrone paths be equal at any equipotential surface, they will be equal at every other. Hence, taking the angles for any equipoten tial surface, cos cos (j> = constant. As an example, suppose a parabola with its vertex upwards to have for directrix the base of an inverted cycloid; these curves evi dently satisfy the above condition, the one being the free path, the other the brachistochrone, for gravity, and the velocities being in each due to the same horizontal line. And it is seen at once that the product of the cosines of the angles which they make with any horizontal straight line which cuts both is a constant whose mag nitude depends on those of the cycloid and parabola, its value being V7/4<z, where I is the latus rectum of the parabola, and a the diameter of the generating circle of the cycloid. Kinetics of a Particle Generally. 153. Here we must content ourselves with a few special Genera cases, which will be varied as much as possible. exampl A unit particle moves on a smooth curve, under the action of any system of forces ; find the motion. All we know directly about the pressure R on the curve is that it is perpendicular to the tangent line at any point. Resolve then the given forces acting upon the particle into three, one, T, along the tangent, which in all cases in nature will be a function of x, y, z and therefore of s ; another, N, in the line of intersection of the normal and osculating planes (or radius of abso lute curvature) ; and the third, P, perpendicular to the osculating plane. Let the resolved parts of R in the directions of N and P be R u R 2 . Now the acceleration of a point moving in any manner is com pounded of two accelerations, one s orv along the tangent to dt* as the path, and the other towards the centre of absolute curva- P ture, the acceleration perpendicular to the osculating plane being
zero ; and therefore