# 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

$$J( x) = \ \int\limits _ { t _ {0} } ^ { {t _ 1 } } L( t, x, \dot{x} ) dt,\ \ L: \mathbf R \times \mathbf R ^ {n} \times \mathbf R ^ {n} \rightarrow \mathbf R ,$$

in the form of a curvilinear integral of the Weierstrass ${\mathcal E}$- function. Let the vector function $x _ {0} ( t)$ be an extremal of the functional $J( x)$, and let it be included in an extremal field with vector-valued field slope function $U( t, x)$ and action $S( t, x)$, corresponding to this field (cf. Hilbert invariant integral). Weierstrass' formula

$$\tag{1 } J( x) = S( t _ {1} , x( t _ {1} ))- S( t _ {0} , x ( t _ {0} )) +$$

$$+ \int\limits _ \gamma {\mathcal E} ( t, x, U( t, x), \dot{x} ) dt$$

applies to any curve $\gamma = x( t)$ in the domain covered by the field. In particular, if the boundary conditions of the curves $\gamma = x( t)$ and $\gamma _ {0} = x _ {0} ( t)$ are identical, i.e. if $x( t _ {i} ) = x _ {0} ( t _ {i} )$, $i= 0, 1$, one obtains Weierstrass' formula for the increment of a functional:

$$\tag{2 } \Delta J = J( x) - J( X _ {0} ) =$$

$$= \ \int\limits _ { t _ {0} } ^ { {t _ 1 } } {\mathcal E} ( t, x( t), U( t, x( t)), \dot{x} ( t)) dt.$$

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=49191
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article