9. Some more examples.

Example 7: Determine the shortest curve y=y(x), y(0)=a, y(1)=b, that spans the area A together with the coordinate axes and the line x=1.


Solution: The problem thus means that we should minimize the functional

under the constraint




By using the method from the previous part we must construct the functional

A necessary condition for minimum is that


fulfills Euler’s equation


Integration now gives

If we integrate again we see that 


that is


which is a circle. The unknown constants  , d0 and c are deterermined from the three conditions


Example 8:
 (Chain line). A rope of length l and constant mass  per unit length hangs between two fixed points (a,y
1) and (b,y2) in the plane. What shape y(x) will the rope attain?

Solution: A small element ds in the point (x,y) has the mass 

and the potential energy 


with respect to y=0. The total potential energy is thus


The rope assumes a shape such that the potential energy is minimized. This means that we should minimize J(y) under the constraint of constant length


We thus construct 


and put it into the Euler equation. However, in this case L* does not explicitly depend on x and we can thus use Beltrami’s identity


that is


If we solve for y’ we get 


This is a separable differential equation. Hence integration gives


We now make the change of variables


and then the left hand side integral transforms to


We multiply with  and then take cosh(.) on both sides of the equality sign and use that it is an even function. We get


If we finally solve for y we get


This means that the shape of a hanging rop is a catenary (or chain line). The constants C, C2 and  are determined by the boundary conditions and the curve length.

Example 9: Find the extremals to the triple integral

under the constraint 


We thus construct


The Euler equation in this case becomes 


that is


This is the famous Schrödinger equation in quantum mechanics for a particle of mass m under the influence of a potential V. In this case D is R3and the constraints is the normalized probability condition for the wave function  , whose square is a probablility density (a measure of where a particle most likely will be). In general we can only find solutions  for particular discrete values of the multiplier  , identified as the possible energy levels that the particla may have.