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").[1]

Conditions

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

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

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.

References

1. Gelfand, I. M.; Fomin, S. V. (1963). Calculus of Variations. Englewood Cliffs, NJ: Prentice-Hall. pp. 61–63.