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. $\left.{\frac {\partial f}{\partial y'}}\right|_{x=c-0}=\left.{\frac {\partial f}{\partial y'}}\right|_{x=c+0}$
2. $\left.\left(f-y'{\frac {\partial f}{\partial y'}}\right)\right|_{x=c-0}=\left.\left(f-y'{\frac {\partial f}{\partial y'}}\right)\right|_{x=c+0}$.

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.