By U. Brechtken-Manderscheid (auth.)
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Example text
Then we will show that for the prescribed boundary values there can be one, two or no extremals. 28 The Euler differential equation \ I \ \ Fig. 1. Catenaries. The integrand here is independent of x. Thus, from the Euler equation we must have Solve for y' to obtain Then separating variables leads to the extremals 1 c y(x) = - cosh(cx + d). Here c > 0 and d are constants of integration. These curves are called catenaries (cf. Fig. 1). ) = (0,1). Also let b = 1. We will not assign a particular value to Yb but rather investigate which values Yb > 0 may have here.
Thus the condition that 4Y'(x)(y 12 (x) - 1) be constant must be satisfied by the extremals. We can conclude that Y'( x) is constant and that Y = CIX + C2. We need to check now whether 36 The Euler differential equation corners can occur. To this end (cf. 29)), we investigate the expressions fyl(X, y, y') = 4y'(y,2 - 1) and f(x,y,y') - fyl(X,y,y')y' = (y'2 -I? _4y'2(y'2 -1) = (y'2 _ 1)( _3y'2 - 1). The corner conditions require that for arbitrary x and y we must have and f(x, y, -1) - fyl(X, y, -1)( -1) = f(x, y, 1) - fyl(X, y, 1)1.
3) there are admissible functions y which are defined piecewise by y(X) = ±x + c, c = constant, which satisfy the corner conditions at their corners and which also satisfy the Euler equation elsewhere. These 'zig-zag' functions are solutions. 27) it follows that if Yo is smooth for Xo E ( a, b), then f~( u) du is differentiable with f: 37 The Erdmann corner conditions Y B' A / B b x Fig. 3. Possible solutions with corners. respect to x at Xo. Therefore, f~1 is also differentiable and we may write (cf.