Mathematical Methods for Physics and Engineering - Matematica.NET

22.3 SOME EXTENSIONSbzρ−ba(a) (b) (c)Figure 22.4Possible soap films between two parallel circular rings.surface area between z and z + dz isso the total surface area is given bydS =2πρ [ (dz) 2 +(dρ) 2] 1/2,S =2π∫ b−bρ(1 + ρ ′2 ) 1/2 dz. (22.11)Since the integrand does not contain z explicitly, we can use (22.8) to obtain an equationfor ρ that minimises S, i.e.ρ(1 + ρ ′2 ) 1/2 − ρρ ′2 (1 + ρ ′2 ) −1/2 = k,where k is a constant. Multiplying through by (1 + ρ ′2 ) 1/2 , rearranging to find an explicitexpression for ρ ′ and integrating we findcosh −1 ρk = z k + c.where c is the constant of integration. Using the boundary conditions ρ(±b) =a, werequire c =0andk such that a/k =coshb/k (if b/a istoolarge,nosuchk can be found).Thus the curve that minimises the surface area isρ/k =cosh(z/k),and in profile the soap film is a catenary (see section 22.4) with the minimum distancefrom the axis equal to k. ◭22.3 Some extensionsIt is quite possible to relax many of the restrictions we have imposed so far. Forexample, we can allow end-points that are constrained to lie on given curves ratherthan being fixed, or we can consider problems with several dependent and/orindependent variables or higher-order derivatives of the dependent variable. Eachof these extensions is now discussed.781