22 Calculus of variations
In chapters 2 and 5 we discussed how to find stationary values of functions of a
single variablef(x), of several variablesf(x,y,...) and of constrained variables,
wherex,y,...are subject to thenconstraintsgi(x,y,...)=0,i=1, 2 ,...,n.Inall
these cases the forms of the functionsfandgiwere known, and the problem was
one of finding the appropriate values of the variablesx,yetc.
We now turn to a different kind of problem in which we are interested inbringing about a particular condition for a given expression (usually maximising
or minimising it) by varying thefunctionson which the expression depends. For
instance, we might want to know in what shape a fixed length of rope should
be arranged so as to enclose the largest possible area, or in what shape it will
hang when suspended under gravity from two fixed points. In each case we are
concerned with a general maximisation or minimisation criterion by which the
functiony(x) that satisfies the given problem may be found.
The calculus of variations provides a method for finding the functiony(x).The problem must first be expressed in a mathematical form, and the form
most commonly applicable to such problems is anintegral.Ineachoftheabove
questions, the quantity that has to be maximised or minimised by an appropriate
choice of the functiony(x) may be expressed as an integral involvingy(x)and
the variables describing the geometry of the situation.
In our example of the rope hanging from two fixed points, we need to findthe shape functiony(x) that minimises the gravitational potential energy of the
rope. Each elementary piece of the rope has a gravitational potential energy
proportional both to its vertical height above an arbitrary zero level and to the
length of the piece. Therefore the total potential energy is given by an integral
for the whole rope of such elementary contributions. The particular functiony(x)
for which the value of this integral is a minimum will give the shape assumed by
the hanging rope.