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
, 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