Necessary Condition for Integral Functional to have Extremum for given Function/Non-differentiable at Intermediate Point

Theorem
Let $y,~ F$ be functions.

Let $ y$ be continuously differentiable for $x \in \left[{{ a}\,.\,.\,{c}} \right) \cap \left({{ c}\,.\,.\,{ b}} \right]$ and satisfy


 * $ y \left({ a} \right)=A,\quad y \left({ b} \right)=B$

Let $J[ y]$ be a functional of the form


 * $ \displaystyle J[ y]= \int_a^b F \left({ x, y, y'} \right) \mathrm d x$

Then the functional $J$ has a weak extremum if $ y$ satisfies the following system of equations:

$ \begin{cases} & \displaystyle F_y- \frac{ \mathrm d }{ \mathrm d x }F_{y'}=0 \\ & \displaystyle \lim_{ x \to c-0} F_{ y'} = \lim_{ x \to c+0} F_{ y'} \\ & \displaystyle \lim_{ x \to c-0} \left({ F - y' F_{ y'} } \right) = \lim_{ x \to c+0} \left({ F - y' F_{ y'} } \right) \end{cases}$

The last two equations are known as the Weierstrass-Erdmann corner conditions.

Proof
Rewrite $J[ y]$ as a sum of two functionals:

Recall that end points $x=a, ~x=b$ are fixed.

The function $y(x)$ has to be $C^0$ at $x=c$, but otherwise this point can move freely.

From general variation of functional, and noting that $y=y\left({ x}\right)$ is an extremal, write down variations for $J_1[ y]$ and $J_2[ y]$ separately:

$\displaystyle \delta J_1 = \lim_{x \to c-0} \left[{ F_{ y'} } \right] \delta y_1+ \lim_{x \to c-0} \left[{ F- y' F_{y'}   } \right] \delta x_1$

$\displaystyle \delta J_2 = -\lim_{x \to c+0} \left[{ F_{ y'} } \right] \delta y_1- \lim_{x \to c+0} \left[{ F- y' F_{y'}   } \right] \delta x_1$

Note that $ \delta J_1$ and $ \delta J_2$ involve the same increments $ \delta x_1$ and $ \delta y_1$.

Since $y=y \left({ x} \right)$ is an extremum of $J$, we have:


 * $ \delta J=0$

or


 * $ \delta J_1 + \delta J_2=0$

Therefore:

Since $ \delta x_1$ and $ \delta y_1$ are arbitrary, both collections of terms have to vanish independently.