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''in the classical calculus of variations''
 
''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
 
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>
+
$$
 +
J(x) \  = \  \int\limits _ {t _ 0} ^ {t _ 1} L(t,\  x,\  \dot{x} ) \  dt,\ \
 +
L : \  \mathbf R \times \mathbf R ^{n} \times \mathbf R ^{n} \rightarrow \mathbf R ,
 +
$$
  
the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w0974404.png" />-function has the form
+
the $  {\mathcal E} $-
 +
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>
+
$$ \tag{1}
 +
{\mathcal E} (t,\  x,\  \dot{x} ,\  x ^ \prime  )\  =
 +
$$
  
<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/w0974406.png" /></td> </tr></table>
+
$$
 +
= \
 +
L ( t,\  x,\  x ^ \prime  ) - L ( t,\  x,\  \dot{x} ) - ( x
 +
^ \prime  - \dot{x} ,\  L _{ {\dot{x} }} ( t,\  x,\  \dot{x} )).
 +
$$
  
 
If one introduces the function
 
If one introduces 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/w0974407.png" /></td> </tr></table>
+
$$
 +
\Pi (t,\  x,\  p,\  x ^ \prime  ) \  = \  (p,\  x ^ \prime  ) - L ( t,\  x,\  x ^ \prime  )
 +
$$
  
(cf. [[Legendre transform|Legendre transform]]; [[Pontryagin maximum principle|Pontryagin maximum principle]]), the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w0974408.png" />-function assumes the form
+
(cf. [[Legendre transform|Legendre transform]]; [[Pontryagin maximum principle|Pontryagin maximum principle]]), the $  {\mathcal E} $-
 +
function assumes 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/w0974409.png" /></td> </tr></table>
+
$$
 +
{\mathcal E} (t,\  x,\  \dot{x} ,\  x ^ \prime  ) \  = \  \Pi ( t,\  x,\  p,\  \dot{x} ) -
 +
\Pi ( t,\  x,\  p,\  x ^ \prime  ),
 +
$$
  
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
+
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
  
<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>
+
$$
 +
\Pi (x ^{*} ,\  x ^ \prime  ) \  = \  \langle  x ^{*} ,\  x ^ \prime  \rangle - f ( x ^ \prime  ) ,
 +
$$
  
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
+
where $  x ^{*} $
 +
is the derivative $  f ^ {\  \prime} (x) $
 +
of $  f $
 +
at $  x $(
 +
or the subdifferential element if $  f $
 +
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>
+
$$
 +
{\mathcal E} ( x,\  x ^ \prime  ,\  x ^{*} ) \  = \  \Pi ( x ^{*} ,\  x ) -
 +
\Pi ( x ^{*} ,\  x ^ \prime  )
 +
$$
  
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,
+
is the $  {\mathcal E} $-
 +
function constructed from $  f $.  
 +
If $  f $
 +
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>
+
$$ \tag{2}
 +
{\mathcal E} ( x ,\  x ^ \prime  ) \  = \  f ( x ^ \prime  ) - f ( x ) - < x ^ \prime  - x ,\  f ^ {\  \prime}
 +
(x) > ,
 +
$$
  
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.
+
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
 
In the case of a 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/w09744034.png" /></td> </tr></table>
+
$$
 +
\int\limits _{T} L (t,\  x,\  \dot{x} ) \  dt,\ \  t \  = \  (t _{1} \dots t _{n} ) ,
 +
\ \  T \  \subset \  \mathbf R ^{m} ,
 +
$$
  
<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/w09744035.png" /></td> </tr></table>
+
$$
 +
L : \  \mathbf R ^{m} \times \mathbf R \times \mathbf R ^{m} \  \rightarrow \  \mathbf R ,\ \  \dot{x} \  = \  \left (
 +
\frac{\partial x}{\partial t _ 1}
 +
\dots
 +
\frac{\partial x}{\partial t _ m}
 +
\right ) ,
 +
$$
  
in a multi-dimensional variational problem, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744036.png" />-function has the following form:
+
in a multi-dimensional variational problem, the $  {\mathcal E} $-
 +
function has the following 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/w09744037.png" /></td> </tr></table>
+
$$
 +
{\mathcal E} ( t,\  x,\  z,\  z ^ \prime  ) \  = \  L (t,\  x,\  z ^ \prime  ) -
 +
L (t,\  x,\  z ) -
 +
$$
  
<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/w09744038.png" /></td> </tr></table>
+
$$
 +
-  
 +
(z ^ \prime  - z ,\  L _{z} ( t, x, z ) ).
 +
$$
  
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
+
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
  
<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/w09744043.png" /></td> </tr></table>
+
$$
 +
\widetilde{L}  \  = \  L + \sum _{i=1} ^ s \lambda _{i} \phi _{i}  $$
  
has been substituted for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097440/w09744044.png" />.
+
has been substituted for $  L $.
  
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]]).
+
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, <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]]].
+
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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 22:37, 28 January 2020


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; 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 with boundaries $ \phi _{i} (t,\ x,\ \dot{x} ) = 0 $ and 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 [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 $ J $ 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, $ {\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 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)

Comments

References

[a1] L. Cesari, "Optimization - Theory and applications" , Springer (1983)
[a2] G.M. Ewing, "Calculus of variations with applications" , Dover, reprint (1985)
[a3] E.B. Lee, L. Marcus, "Foundations of optimal control theory" , Wiley (1967)
How to Cite This Entry:
Weierstrass E-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_E-function&oldid=19110
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article