Currently I am working on an optimization of the induced drag of lifting objects. In the original paper by Munk(1919) the optimization is performed for an arbitrary distribution of lifting elements throughout a three dimensional space. Then a single constraint of a finite total lift force is imposed which leads to the conclusion that the induced velocities must be constant along every lifting element.
$$ \begin{align}\label{eq:chap4:opt1} \frac{\mathrm{d}J(f(\cdot) + \sigma \delta f(\cdot)}{\mathrm{d}\sigma}\bigg{|}_{\sigma=0} = \frac{2}{V_\infty} \int\limits_h \int\limits_l \delta f(\xi, \zeta) (w - \frac{V_\infty \lambda_1}{2}) \mathrm{d}\xi \mathrm{d}\zeta + 2 \int\limits_h \int\limits_l \delta F(\xi, \zeta) u \mathrm{d}\xi \mathrm{d}\zeta = 0 \notag\\ \end{align} $$
Here the goal function J is a funtion of f, and F. The $\delta$ means an arbitrary variation of these functions. Therefore the only manner in which this problem can be equal to 0 is if
$$ w = \frac{V_\infty \lambda_1}{2} $$ and $$ u = 0 $$
The issue arises when another constraint is added. Instead of the situation where there is an arbitrary distribution of lifting elements throughout space the situation is analyzed where there exist so called lifting lines. One aspect of these representations of wings is that they can only create a lift force perpendicular to it's own tangent. Thus the second constraint arises where
$$ f(x,z) \sin{\beta} - F(x,z) \cos{\beta} = 0 $$
This determines that along the line no force exists in the direction of the line itself. When this constraint is imposed on the optimization problem we find:
$$ \begin{align} \frac{\mathrm{d}J(f(\cdot) + \sigma \delta f(\cdot)}{\mathrm{d}\sigma}\bigg{|}_{\sigma=0} = \frac{2}{V_\infty} \int\limits_h \int\limits_l \delta f(\xi, \zeta) (w - \frac{V_\infty \lambda_1}{2}) \mathrm{d}\xi \mathrm{d}\zeta + 2 \int\limits_h \int\limits_l \delta F(\xi, \zeta) u \mathrm{d}\xi \mathrm{d}\zeta \notag\\ - \lambda_2 \left( \delta f(x,z) \sin{\beta} + \delta F(x,z) \cos{\beta} \right) = 0 \end{align} $$
Or when we determine that the term
$$ \left( \delta f(x,z) \sin{\beta} + \delta F(x,z) \cos{\beta} \right) $$
is equal to zero along the entire domain
$$ \begin{equation} \lambda_2 \left( \delta f(x,z) \sin{\beta} - \delta F(x,z) \cos{\beta} \right) = \int\limits_h \int\limits_l \lambda_2 \left( \delta f(x,z) \sin{\beta} - \delta F(x,z) \cos{\beta} \right) \mathrm{d}\xi \mathrm{d}\zeta = 0 \end{equation} $$
Then dividing by $cos(\beta) sin(\beta)$ and inserting this into the original relation we find
$$ \begin{equation} \frac{\mathrm{d}J(f(\cdot) + \sigma \delta f(\cdot)}{\mathrm{d}\sigma}\bigg{|}_{\sigma=0} = \frac{2}{V_\infty} \int\limits_h \int\limits_l \delta f(\xi, \zeta) (w - \frac{V_\infty \lambda_1}{2} - \frac{V_\infty \lambda_2}{2 \cos{\beta}}) \mathrm{d}\xi \mathrm{d}\zeta + \frac{2}{V_\infty} \int\limits_h \int\limits_l \delta F(\xi, \zeta) (u + \frac{V_\infty \lambda_2}{2 \sin{\beta}} ) \mathrm{d}\xi \mathrm{d}\zeta =0 \end{equation} $$
Thus two equations are found: $$ \begin{equation} w - \frac{V_\infty \lambda_1}{2} - \frac{V_\infty \lambda_2}{2 \cos{\beta}} = 0 \end{equation} $$ And $$ \begin{equation} u + \frac{V_\infty \lambda_2}{2 \sin{\beta}} = 0 \end{equation} $$ Thus $$ \begin{equation} V_\infty \lambda_2 = - u 2 \sin{\beta} \end{equation} $$ Then multiplying by $\cos{\beta}$ and inserting $\lambda_2$ it is found: $$ \begin{equation} w \cos{\beta} + u \sin{\beta} = \frac{V_\infty \lambda_1}{2} \cos{\beta} \end{equation} $$
For which the original solution for a constant $w = \frac{V_\infty \lambda_1}{2}$ is also part of the solution space. But as far as I know the adding of a constraint to an optimization problem should not lead to an increase of the solution space (here u is allowed to have a value).
So what am I missing or doing wrong?
I know that the division by $cos(\beta)sin(\beta)$ causes some problems when $\beta= 0; \pi/2$ but the conclusions of this paper have been used for 100 years and they seem to be completely correct.