Weierstrass formula

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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


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:


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


[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:
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article