Weierstrass–Erdmann condition

The Weierstrass–Erdmann condition is a mathematical result from the calculus of variations, which specifies sufficient conditions for broken extremals (that is, an extremal which is constrained to be smooth except at a finite number of "corners").

Conditions
The Weierstrass-Erdmann corner conditions stipulate that a broken extremal $$y(x)$$ of a functional $$J=\int\limits_a^b f(x,y,y')\,dx$$ satisfies the following two continuity relations at each corner $$c\in[a,b]$$:

1. $\left.\frac{\partial f}{\partial y'}\right

2. _{x=c-0}=\left.\frac{\partial f}{\partial y'}\right

3. _{x=c+0}$

4. $\left.\left(f-y'\frac{\partial f}{\partial y'}\right)\right

5. _{x=c-0}=\left.\left(f-y'\frac{\partial f}{\partial y'}\right)\right

6. _{x=c+0}$.

Applications
The condition allows one to prove that a corner exists along a given extremal. As a result, there are many applications to differential geometry. In calculations of the Weierstrass E-Function, it is often helpful to find where corners exist along the curves. Similarly, the condition allows for one to find a minimizing curve for a given integral.