# 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 where 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.