View source for Weierstrass E-function

''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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w0974403.png" /></td> </tr></table>

the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w0974404.png" />-function has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w0974405.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

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If one introduces the function

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(cf. [[Legendre transform|Legendre transform]]; [[Pontryagin maximum principle|Pontryagin maximum principle]]), the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w0974408.png" />-function assumes the form

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744010.png" />. The general construction of functions analogous to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744011.png" />-function (1) consists of the following. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744012.png" /> be a differentiable or convex function, defined on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744013.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744014.png" /> be the dual space. If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744015.png" /> is defined by the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744016.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744017.png" /> is the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744019.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744020.png" /> (or the subdifferential element if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744021.png" /> is convex), the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744022.png" /></td> </tr></table>

is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744023.png" />-function constructed from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744024.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744025.png" /> is differentiable,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

i.e. the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744027.png" />-function is the difference at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744028.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744029.png" /> and the linear function tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744030.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744031.png" />. A comparison of formulas (1) and (2) shows that in the classical calculus of variations the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744032.png" />-function is obtained from the construction (2) with respect to the variables related to derivatives, while the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744033.png" /> play the role of parameters.

In the case of a functional

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in a multi-dimensional variational problem, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744036.png" />-function has the following form:

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In the case of the [[Lagrange problem|Lagrange problem]] with boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744039.png" /> and [[Lagrange multipliers|Lagrange multipliers]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744041.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744042.png" />-function has the form (1), in which

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has been substituted for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744044.png" />.

The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744045.png" />-function, first introduced in 1879 by K. Weierstrass [[#References|[1]]], lies at the foundation of the theory of the calculus of variations (cf. [[Variational calculus|Variational calculus]]). It is used in the formulation of necessary and (partially) sufficient conditions for an extremum (cf. [[Weierstrass conditions (for a variational extremum)|Weierstrass conditions (for a variational extremum)]]), and serves to express the increment of a functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744046.png" /> on an extremal in the form of a finite integral (cf. [[Weierstrass formula|Weierstrass formula]]).

An especially important role in variational calculus is played by smooth functionals in which, in a given parameter range, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744047.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744048.png" />, or, stronger, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744049.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744050.png" />. They are known as quasi-regular and regular or elliptic, respectively. For such functionals the [[Legendre condition|Legendre condition]] and the necessary [[Weierstrass conditions (for a variational extremum)|Weierstrass conditions (for a variational extremum)]] are invariably valid, as are the theorems of existence and regularity [[#References|[7]]].

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



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Cesari,   "Optimization - Theory and applications" , Springer  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.M. Ewing,   "Calculus of variations with applications" , Dover, reprint  (1985)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.B. Lee,   L. Marcus,   "Foundations of optimal control theory" , Wiley  (1967)</TD></TR></table>