# CoV23

From Example Problems

Find the extremals of the functional subject to the constraint

Let where is
a Lagrange multiplier. The Euler equation is

(i) Try ; let . Then

But the solution for this is , i.e., which does not satisfy the constraint

(ii) Try Then

so ; then
which, again, does not satisfy the constraint.

(iii) ; let . Then

; or

, so we reject this and take ,
which implies

Thus where is any nonzero integer.

is found by imposing the constraint:

Hence, the extremals of subject to the given constraint are