# Weierstrass formula

*for the increment of a functional*

A formula in the classical calculus of variations (cf. Variational calculus), defining the values of the functional

in the form of a curvilinear integral of the Weierstrass -function. Let the vector function be an extremal of the functional , and let it be included in an extremal field with vector-valued field slope function and action , corresponding to this field (cf. Hilbert invariant integral). Weierstrass' formula

(1) |

applies to any curve in the domain covered by the field. In particular, if the boundary conditions of the curves and are identical, i.e. if , , one obtains Weierstrass' formula for the increment of a functional:

(2) |

Formulas (1) and (2) are sometimes referred to as Weierstrass' fundamental theorem.

#### References

[1] | C. Carathéodory, "Calculus of variations and partial differential equations of the first order" , 1–2 , Holden-Day (1965–1967) (Translated from German) |

[2] | L. Young, "Lectures on the calculus of variations and optimal control theory" , Saunders (1969) |

[3] | N.I. Akhiezer, "The calculus of variations" , Blaisdell (1962) (Translated from Russian) |

**How to Cite This Entry:**

Weierstrass formula.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_formula&oldid=16280