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

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

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)