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 [math]\displaystyle{ y(x) }[/math] of a functional [math]\displaystyle{ J=\int\limits_a^b f(x,y,y')\,dx }[/math] satisfies the following two continuity relations at each corner [math]\displaystyle{ c\in[a,b] }[/math]:

1. [math]\displaystyle{ \left.\frac{\partial f}{\partial y'}\right|_{x=c-0}=\left.\frac{\partial f}{\partial y'}\right|_{x=c+0} }[/math]
2. [math]\displaystyle{ \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} }[/math].

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.