Weierstrass E-function

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in the classical calculus of variations

A function which isolates the main part of the increment of a functional as the extremal is varied, using a local (needle-shaped) variation for a given value of its derivative, at a given point of the extremal. In the case of the functional

the -function has the form

 (1)

If one introduces the function

(cf. Legendre transform; Pontryagin maximum principle), the -function assumes the form

where . The general construction of functions analogous to the -function (1) consists of the following. Let be a differentiable or convex function, defined on a Banach space , and let be the dual space. If the function is defined by the equation

where is the derivative of at (or the subdifferential element if is convex), the function

is the -function constructed from . If is differentiable,

 (2)

i.e. the -function is the difference at between and the linear function tangent to at . A comparison of formulas (1) and (2) shows that in the classical calculus of variations the -function is obtained from the construction (2) with respect to the variables related to derivatives, while the variables play the role of parameters.

In the case of a functional

in a multi-dimensional variational problem, the -function has the following form:

In the case of the Lagrange problem with boundaries and Lagrange multipliers , , the -function has the form (1), in which

has been substituted for .

The -function, first introduced in 1879 by K. Weierstrass [1], lies at the foundation of the theory of the calculus of variations (cf. Variational calculus). It is used in the formulation of necessary and (partially) sufficient conditions for an extremum (cf. Weierstrass conditions (for a variational extremum)), and serves to express the increment of a functional on an extremal in the form of a finite integral (cf. Weierstrass formula).

An especially important role in variational calculus is played by smooth functionals in which, in a given parameter range, for all , or, stronger, if for all . They are known as quasi-regular and regular or elliptic, respectively. For such functionals the Legendre condition and the necessary Weierstrass conditions (for a variational extremum) are invariably valid, as are the theorems of existence and regularity [7].

References

 [1] K. Weierstrass, "Vorlesungen über Variationsrechnung" , Math. Werke , 7 , Akademie Verlag (1927) [2] C. Carathéodory, "Calculus of variations and partial differential equations of the first order" , 1–2 , Holden-Day (1965–1967) (Translated from German) [3] O. Bolza, "Lectures on the calculus of variations" , Chelsea, reprint (1960) (Translated from German) [4] N.I. Akhiezer, "The calculus of variations" , Blaisdell (1962) (Translated from Russian) [5] L.S. Pontryagin, V.G. Boltayanskii, R.V. Gamkrelidze, E.F. Mishchenko, "The mathematical theory of optimal processes" , Wiley (1962) (Translated from Russian) [6] M.R. Hestenes, "Calculus of variations and optimal control theory" , Wiley (1966) [7] "Hilbert problems" Bull. Amer. Math. Soc. , 8 (1902) pp. 437–479 (Translated from German) [8] G.A. Bliss, "Lectures on the calculus of variations" , Chicago Univ. Press (1947)