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

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The Weierstrass-Erdmann corner conditions stipulate that a broken extremal   of a functional   satisfies the following two continuity relations at each corner  :

  1.  
  2.  .

Applications

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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

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  1. ^ Gelfand, I. M.; Fomin, S. V. (1963). Calculus of Variations. Englewood Cliffs, NJ: Prentice-Hall. pp. 61–63. ISBN 9780486135014.