Lectures on the Calculus of Variations
by Oskar Bolza
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from ... Show synopsis This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1904 Excerpt: ...and y' we get x'Fx.x. + y'Fy.x. = 0, x'Fx.y. + y'F, .y. = 0; hence if x' and y' are not both zero, Fx.x.: Fx.y.: Fy.y. = y" -x'y': x'2; (11) there exists therefore a function 1F of x, y, x, y' such that F-x. = y'2Ft, Fx.y. =-x'y'F, Fy.y. = x"F, . (Ha) The function Fx thus defined is of class C in the domain (r), even when one of the two variables x', y' is zero; but Fl becomes in general infinite when x' and y vanish simultaneously, even if F itself should remain finite and continuous for x'--O, y' = 0. For instance: F = yV/x" + y," F1=.. (l x 2 + y 2) c) Definition of a Minimum: Two points A(x0, y0) and B(xu?/i) being given in the region K, we consider the totality M of all ordinary2 curves which can be drawn in 1R from A to B. Then a curve 6 of M is said to minimize the integral J= I F(x, y, x, y')dt, if there exists a neighborhood H of 6 such that JJ (12) for every ordinary curve 6 which can be drawn in H from A to B. We may, without loss of generality, choose for H the strip3 of the x, ?/-plane swept over by a circle of constant radius p whose center moves along the curve 6 from A to B. This strip will be called "the neighborhood (p) of 6." Compare 3. The definition is due to Weierstrass, Lectures, 1879; compare also Zerhelo, Dissertation, pp. 25-29, and Kneser, Lehrbuch, 17. 2 An extension of the problem to a still more general class of curves will be considered in 31. 3 In case different portions of the strip should overlap, the plane has to be imagined as multiply covered in the manner of a Riemanu-surf ace (weierstrass). h being the second constant of integration. The extremals are therefore cycloids' generated by a circle of radius r rolling upon the horizontal line y--y0+k = 0. To these two equ.