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Weierstrass–Erdmann condition

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

  1. ∂ f ∂ y ′ | x = c − 0 = ∂ f ∂ y ′ | x = c + 0 {\displaystyle \left.{\frac {\partial f}{\partial y'}}\right|_{x=c-0}=\left.{\frac {\partial f}{\partial y'}}\right|_{x=c+0}}
  2. ( f − y ′ ∂ f ∂ y ′ ) | x = c − 0 = ( f − y ′ ∂ f ∂ y ′ ) | x = c + 0 {\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. ISBN 9780486135014.
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