Mathematics for Economists

Calculus of variations

Example

Find the extremal of the functional

Z 2 π

0

x^2 (t)x^2 (t)dt, x( 0 )=x( 2 π)= 1.

The kernel function is

F



t,x,x



= x

2

x^2.

Fx^0



t,x,x



=  2 x,Fx^0



t,x,x



= 2 x.

The EulerñLagrange equation is

 2 x= d

dt

2 x,)x+x= 0.

which has the general solutionx(t)=c 1 cost+c 2 sint.Putting the

boundary conditions the extremal solutions are of the form

x(t)=cost+csint.