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
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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) |
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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) |
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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) |
Weierstrass formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_formula&oldid=16280