Mathematical Methods for Physics and Engineering - Matematica.NET

CALCULUS OF VARIATIONSwe can use (22.8) to obtain a first integral of the EL equation for y, namelyy(1 − y ′2 ) 1/2 + yy ′2 (1 − y ′2 ) −1/2 = k,where k is a constant. On rearranging this givesky ′ = ±(k 2 − y 2 ) 1/2 ,which, using y(0) = 0, integrates toy/k = sin(s/k). (22.10)The other end-point, y(l/2) = 0, fixes the value of k as l/(2π) to yieldy =l2π2πssin .lFrom this we obtain dy =cos(2πs/l) ds and since (ds) 2 =(dx) 2 +(dy) 2 we find also thatdx = ± sin(2πs/l) ds. This in turn can be integrated and, using x(0) = 0, gives x in termsof s asx − l2π = − l2π2πscos .lWe thus obtain the expected result that x and y lie on the circle of radius l/(2π) givenby(x − l ) 2+ y 2 = l22π 4π . 2Substituting the solution (22.10) into the expression for the total area (22.9), it is easilyverified that A = l 2 /(4π). A much quicker derivation of this result is possible using planepolar coordinates. ◭The previous two examples have been carried out in some detail, even thoughthe answers are more easily obtained in other ways, expressly so that the methodis transparent and the way in which it works can be filled in mentally at almostevery step. The next example, however, does not have such an intuitively obvioussolution.◮Two rings, each of radius a, are placed parallel with their centres 2b apart and on acommon normal. An open-ended axially symmetric soap film is formed between them (seefigure 22.4). Find the shape assumed by the film.Creating the soap film requires an energy γ per unit area (numerically equal to the surfacetension of the soap solution). So the stable shape of the soap film, i.e. the one thatminimises the energy, will also be the one that minimises the surface area (neglectinggravitational effects).It is obvious that any convex surface, shaped such as that shown as the broken line infigure 22.4(a), cannot be a minimum but it is not clear whether some shape intermediatebetween the cylinder shown by solid lines in (a), with area 4πab (or twice this for thedouble surface of the film), and the form shown in (b), with area approximately 2πa 2 , willproduce a lower total area than both of these extremes. If there is such a shape (e.g. that infigure 22.4(c)), then it will be that which is the best compromise between two requirements,the need to minimise the ring-to-ring distance measured on the film surface (a) and theneed to minimise the average waist measurement of the surface (b).We take cylindrical polar coordinates as in figure 22.4(c) and let the radius of the soapfilm at height z be ρ(z) withρ(±b) =a. Counting only one side of the film, the element of780