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

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

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

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

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