Mathematical Methods for Physics and Engineering - Matematica.NET

22Calculus of variationsIn chapters 2 and 5 we discussed how to find stationary values of functions of asingle variable f(x), of several variables f(x,y,...) and of constrained variables,where x,y,... are subject to the n constraints g i (x,y,...)=0,i =1, 2,...,n.Inallthese cases the forms of the functions f and g i were known, and the problem wasone of finding the appropriate values of the variables x, y etc.We now turn to a different kind of problem in which we are interested inbringing about a particular condition for a given expression (usually maximisingor minimising it) by varying the functions on which the expression depends. Forinstance, we might want to know in what shape a fixed length of rope shouldbe arranged so as to enclose the largest possible area, or in what shape it willhang when suspended under gravity from two fixed points. In each case we areconcerned with a general maximisation or minimisation criterion by which thefunction y(x) that satisfies the given problem may be found.The calculus of variations provides a method for finding the function y(x).The problem must first be expressed in a mathematical form, and the formmost commonly applicable to such problems is an integral. Ineachoftheabovequestions, the quantity that has to be maximised or minimised by an appropriatechoice of the function y(x) may be expressed as an integral involving y(x) andthe variables describing the geometry of the situation.In our example of the rope hanging from two fixed points, we need to findthe shape function y(x) that minimises the gravitational potential energy of therope. Each elementary piece of the rope has a gravitational potential energyproportional both to its vertical height above an arbitrary zero level and to thelength of the piece. Therefore the total potential energy is given by an integralfor the whole rope of such elementary contributions. The particular function y(x)for which the value of this integral is a minimum will give the shape assumed bythe hanging rope.775