View source for 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

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

the  $  {\mathcal E} $-
function has the form

$$ \tag{1}
{\mathcal E} (t,\  x,\  \dot{x} ,\  x ^ \prime  )\  =
$$

$$ 
= \ 
L ( t,\  x,\  x ^ \prime  ) - L ( t,\  x,\  \dot{x} ) - ( x
^ \prime  - \dot{x} ,\  L _{ {\dot{x} }} ( t,\  x,\  \dot{x} )).
$$

If one introduces the function

$$ 
\Pi (t,\  x,\  p,\  x ^ \prime  ) \  = \  (p,\  x ^ \prime  ) - L ( t,\  x,\  x ^ \prime  )
$$

(cf. [[Legendre transform|Legendre transform]]; [[Pontryagin maximum principle|Pontryagin maximum principle]]), the  $  {\mathcal E} $-
function assumes the form

$$ 
{\mathcal E} (t,\  x,\  \dot{x} ,\  x ^ \prime  ) \  = \  \Pi ( t,\  x,\  p,\  \dot{x} ) -
\Pi ( t,\  x,\  p,\  x ^ \prime  ),
$$

where  $  p= L _{ {\dot{x} }} (t,\  x,\  \dot{x} ) $. 
The general construction of functions analogous to the  $  {\mathcal E} $-
function (1) consists of the following. Let  $  f $
be a differentiable or convex function, defined on a Banach space  $  X $, 
and let  $  X ^{*} $
be the dual space. If the function  $  \Pi : \  X ^{*} \times X \rightarrow \mathbf R $
is defined by the equation

$$ 
\Pi (x ^{*} ,\  x ^ \prime  ) \  = \  \langle  x ^{*} ,\  x ^ \prime  \rangle - f ( x ^ \prime  ) ,
$$

where  $  x ^{*} $
is the derivative  $  f ^ {\  \prime} (x) $
of  $  f $
at  $  x $(
or the subdifferential element if  $  f $
is convex), the function

$$ 
{\mathcal E} ( x,\  x ^ \prime  ,\  x ^{*} ) \  = \  \Pi ( x ^{*} ,\  x ) -
\Pi ( x ^{*} ,\  x ^ \prime  )
$$

is the  $  {\mathcal E} $-
function constructed from  $  f $. 
If  $  f $
is differentiable,

$$ \tag{2}
{\mathcal E} ( x ,\  x ^ \prime  ) \  = \  f ( x ^ \prime  ) - f ( x ) - < x ^ \prime  - x ,\  f ^ {\  \prime}
(x) > ,
$$

i.e. the  $  {\mathcal E} $-
function is the difference at  $  x ^ \prime  $
between  $  f $
and the linear function tangent to  $  f $
at  $  x $. 
A comparison of formulas (1) and (2) shows that in the classical calculus of variations the  $  {\mathcal E} $-
function is obtained from the construction (2) with respect to the variables related to derivatives, while the variables  $  t,\  x $
play the role of parameters.

In the case of a functional

$$ 
\int\limits _{T} L (t,\  x,\  \dot{x} ) \  dt,\ \  t \  = \  (t _{1} \dots t _{n} ) ,
\ \  T \  \subset \  \mathbf R ^{m} ,
$$

$$ 
L : \  \mathbf R ^{m} \times \mathbf R \times \mathbf R ^{m} \  \rightarrow \  \mathbf R ,\ \  \dot{x} \  = \  \left ( 
\frac{\partial x}{\partial t _ 1}
 \dots 
\frac{\partial x}{\partial t _ m}
 \right ) ,
$$

in a multi-dimensional variational problem, the  $  {\mathcal E} $-
function has the following form:

$$ 
{\mathcal E} ( t,\  x,\  z,\  z ^ \prime  ) \  = \  L (t,\  x,\  z ^ \prime  ) -
L (t,\  x,\  z ) -
$$

$$ 
- 
(z ^ \prime  - z ,\  L _{z} ( t, x, z ) ).
$$

In the case of the [[Lagrange problem|Lagrange problem]] with boundaries  $  \phi _{i} (t,\  x,\  \dot{x} ) = 0 $
and [[Lagrange multipliers|Lagrange multipliers]]  $  \lambda _{i} (t) $, 
$  i=1 \dots s $, 
the  $  {\mathcal E} $-
function has the form (1), in which

$$ 
\widetilde{L}  \  = \  L + \sum _{i=1} ^ s \lambda _{i} \phi _{i}  $$

has been substituted for  $  L $.

The  $  {\mathcal E} $-
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  $  J $
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,  $  {\mathcal E} ( \cdot ,\  \dot{x} ,\  x ^ \prime  ) \geq 0 $
for all  $  \dot{x} ,\  x ^ \prime  $, 
or, stronger, if  $  {\mathcal E} ( \cdot ,\  \dot{x} ,\  x ^ \prime  ) >0 $
for all  $  \dot{x} \neq x ^ \prime  $. 
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>