A large number of special cases lead to dif ferential equations which can be solved directly.
---- For example, if -1-y"dx, we shall have f(x, y, y')=N/ 1 -Fy'l whence (1 1 , 1 / ( 1 y')., and the equation (6) takes the form y"--o. The only solutions of this differential equation are the straight lines y=ax+ b. It follows that if there is any curve of the class B in the plane along which the distance between two given fixed points is at a minimum, that curve is the straight line joining the two points. This result is independent of the Euclidean postulate, and depends only upon the definition of length by means of the preceding integral.
The problem of the brachistocrone, men tioned above, is to find the curve along which a particle with initial velocity vo will descend most quickly from a given initial point Po to another given point Pi. It is easy to show that the time of descent is given by the formula 'V Hei dx , zo hence Euler's equation (6) is d \_ (1,t dx\ayi dx \V 1 +ye)/ which gives at once (x.—x)). This equation may be readily solved in parameter form, and we find: x — A + B (1—cos 6)), 2 y —C A + B (0.—sin 2 where A= -- + Xe, 2 ct—v°2 xo..
2g These extremals are cycloids on horizontal bases, the radius of the generating circle being (A+B)/2, and one cusp being at the point (A, C). Further investigation is necessary to decide just when a given pair of points can be connected by such a cycloid (cf. Bolza, (Lec tures,' p. 236). If such a cycloid can be drawn, we can infer that it is the solution if there is any solution. If no such cycloid can be drawn, we can infer that there is no solution in the region R.
The problem of finding the geodetic lines on a given surface is that of minimizing the integral, I= xo where s• ==.(x,y) is the surface and where E = F Euler's equa tion therefore coincides with the usual equation for the geodetic lines: d Gy' F _ + 2 Fy y' (ix\ E+2Fy' and the geodetic lines are the extremals of this problem, i.e., no line not a geodetic can be a shortest line on a surface.
Though the proof of the necessity of Euler's condition was satisfactory, even in a cruder form, to the originators of the subject, a desire to formulate sufficient conditions arose. Thus
Legendre showed that a second necessary con dition for a minimum [maximum] is that the condition )>0[ < 01 for be satisfied along the supposed solution between the end points. We shall prove this, and we shall see that the same condition is actually a sufficient condition for a weak lim ited minimum if the sign = be removed.
Jacobi then showed, by means of the second variation of the given integral, that a third necessary condition for a minimum [maximum] is that the quantity d(x, x0)1(x)vo(x0)--ito(x)vi(x4) should not vanish for any value of x in the interval xo< x
It was long believed that Jacobi's condition, together with the previous two, was a sufficient condition. That such is not the case was first pointed out by Weierstrass, who also showed that Jacobi's condition, while not sufficient for a minimum in general, is sufficient for a weak minimum (if the point (x,, 3%) lies inside the envelope of the extremals through (xo, yo). Cf. Bolza, 'Lectures,' Chap. 3.
That the preceding conditions are not suffi cient is most readily seen by giving an actual example in which the extremals, though all the above conditions are satisfied, do not minimize the integral. Such is the example (sec Bolza, (Lectures,' p. 73), f(x, Here the extremals are straight lines, but it is easy to join two points for which all the preceding conditions are satisfied by a simple broken line for which the value of the integral is less than that along the straight line extremal. Of course, the comparison line used varies con siderably from the straight line extremal in direction, though not in position.