This example is valuable in emphasizing the restrictions that surround the derivation and the meaning of the stationary condition. Exercises 7 and 8 exam- ine the conditions for the pathological behavior for a symmetric example. More information can be found in many texts on the calculus of variations. 3. The brachistochrone problem. (See Fig. 2.4a.) This well-known problem is to find the curve joining two points, along which a particle falling from rest under the influence of gravity travels from the higher to the lower point in the least time. If v is the speed along the curve, then the time required to fall an arc length ds is ds/v, and the problem is to find a minimum of the integral or and / is identified as 1 FIGURE 2.4a The brachistochrone problem. 2.2 Some Techniques of the Calculus of Variations If y is measured down from the initial point of release, the conservation theorem for the energy of the particle can be written as m=mgy -f4/5. 712 ==> Then the expression for f12 becomes d 2a 3a 10 v = 2gy. 112== -S 2gy 1+ y2 2gy The integration of Eq. (2.11) with this form for f is straightforward and is left as an exercise. The parametric solution in terms of its one parameter, a. given by x = a(6-sino). y = a(1-cos). is sketched in Fig. 2.4b for the first cycle (0 x2ra) and the beginning of the second cycle. Three cases of solutions are indicated. A power-series expansion of the solution for the limit y < a gives dx, The brachistochrone problem is famous in the history of mathematics, for it was the analysis of this problem by John Bernoulli that led to the formal foundation of the calculus of variations. 9 y = a(x/a). x = 3/ 43 20 Xx