86 Hence, observing that dx dx 2 dx we get df7= VARIATIONS .(8). In general, if V be a function of x, y, (/, y, . . . y^" we have f x i I Xl SU= I (Pdy + P 1 Ddy+ . . . +P,,D"d>j)dx+ / J dx. J J a Now let A denote the differentiation of $y, and D-, the differentia tion of Pj, PZ, &c. ; we may write P m D m Sy - ( - l) m 8yD m P m = (I) - ( - ir/W^tf r x r x id m p Hence j P m D m Sydx=( - l) m | ( ^ 8> J cb;
Applying this to the different terms in the value of 5 U given above, We get This result may be written dP, d P* - dP,, d P . 8. This expression consists of three parts, (1) the de finite integral I (P)8ydj: ) which depends on Sy, the change J x in the form of the function y ; (2) the expression / Vdx, I ^o which depends solely on the change in the limiting values of x: and (3) the quantity /* {(P^y + (P 2 )fy + . . . } , _ / a-o which depends on the variations of y, y, <fec., at the limits. It is often convenient to Avrite (10) in the abbreviated form dU=L+ (11). 9. The principal applications of the calculus of variations have reference to the determination of the form of one or more unknown functions contained in a definite integral, in such a manner that the integral shall have a maximum or minimum value. For instance, to determine the form of the function y which renders U= I Vdx a maximum or J XD - TI a minimum, we have as above SU=L+ I MSydx. Here, as in the differential calculus (see INFINITESIMAL CAL CULUS, 64), for a maximum or minimum value of U it is readily seen that we must have 817=0; this leads to L and I JfSydx = 0. Now the latter integral cannot be zero for all indefinitely small values of 8y unless Hf=0 for all values within the limits of integration. Hence we get the differential equation in y, 10. We here suppose that there is no restriction on Si/, so that for any value of x the increments + 8y and - 8i/ are equally compatible with the conditions of the problem. The reasoning consequently will not apply if the conditions render this impossible. For instance, if a curve be re stricted to lie within a given boundary, then for all points on the boundary the displacements must be inwards, and the opposite displacements are impossible. In this case J the curve satisfying a required maximum or minimum con dition consists partly of portions of the boundary and partly of portions of a curve satisfying the equation M = 0. 11. The equation M=G is iu general a clillbrential equation of the degree 2n ; accordingly its solution usually contains 2n arbitrary con stants. The values of these constants are to be determined by aid of the equations deduced from L = Q in combination with the given conditions at the limits. For example, if V be a function of x, y. and ii solely, the solution of P-- , , 1 = is in general of the form y = f(x, c 1; c 2 ). Now, suppose that the limiting points are re stricted to the curves 2/o=/o(^o) an( ^ 2/i = /i( r i)> then we get at the limits 5y + y () dx =f l) (x )dx and dy 1 + & 1 dx l =f 1 (sc 1 )dx l . Hence, substituting in L = Q the values of dy and dy l derived from these equations, we get, since d.r and dx 1 are arbitrary, We have therefore six equations for the determination of the .six quantities a , ?/ , a^, y lt c, and c%. The integration of the equation J/=0 is much simplified in particular cases. ( ip (1) Let Fbe a function of x and y solely, then we liave , = 0, therefore P 1 = c .............................. (14). (2) If Tbe a function of y and & solely, P=~ also ~ ( H = Py + Prf dP = -yJy + Pj?/. Hence, integrating, we get where c is an arbitrary constant. 12. For example, let us consider Bernoulli s problem of the curve of quickest descent under the action of gravity. Take the axis of r vertical and that of y horizontal, and suppose the particle to start from the point a , 7/ , with the velocity due to the height h ; then, if r be the velocity at any point, we have v- = 2y(x + h-x ), also /x + h - write U= . ; and by (14) we get Now, writing a for 1/c 2 and assuming x + Jt . -,r = a . sin 5 0, we find cfy = tan 0dx = (1 - cos20)rf0 ; therefore y = a(0 - sin 6 cos 6) + K, and we infer that the curve is a cycloid. (a) If we suppose the upper limit to be fixed and the lower re stricted to the curve i/i^/il^i); we get from (13) l + tan0 J / 1 ? (a: 1 )=0 (16); accordingly the curve intersects the bounding curve orthogonally. (/8) Next let the upper limit be restricted to the curve ?/ =_/"( a;,,) ; then since V contains a , we have an additional term in d U arising from the change in .->, viz., d. ^l -,-dx; consequently the coefficient of c^dr dx in SUis I -j-_-dx - { J r + (P a ) C/i Vo) ~ #o)} > and we have I -, -dx=J + (P l ) l) (f () (x () ) - i/ ] (17). J ,T,, a u If we substitute for V its value, we get without difficulty from this equation l + tan6 1 f (x ) = Q. Comparing this with (16), we see that the tangents to the bounding curves at the extremities of the trajectory are parallel (Moigno, op. cit., p. 230). 13. More generally, consider the curve for which U= / fj.ds m _ J .>() = I /"Vl +ydx is a maximum or a minimum, where /j. is a given function of x and >/. Here P= -j-/l + y 2 , P,= 7^- , and we get r ~, I ; hence, if p be the radius of curvature, we have 1 I/ d/j. n du. - cos a T - +cosa y (18), p /JL dx ay) where a, /3 are the angles which the normal makes with the co- 1 For a full discussion of such cases, see Todlumter, On the Calculus
of Variations, 1871 (Adams prize essay), Cambridge.