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[edit]
The Weierstrass-Erdmann corner conditions stipulate that a broken extremal of a functional satisfies the following two continuity relations at each corner :
![{\displaystyle c\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/997256364b06acf0710e5d24da39e8c42991a249)
- .
Applications[edit]
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[edit]
- ^ Gelfand, I. M.; Fomin, S. V. (1963). Calculus of Variations. Englewood Cliffs, NJ: Prentice-Hall. pp. 61–63. ISBN 9780486135014.