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Necessary conditions for an extremum, additional to the [[Euler equation|Euler equation]], specified at points at which the extremal has a corner. Let
 
Necessary conditions for an extremum, additional to the [[Euler equation|Euler equation]], specified at points at which the extremal has a corner. Let
  
<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/w097460/w0974601.png" /></td> </tr></table>
+
$$
 +
J ( x)  = \int\limits L ( t, x, \dot{x} )  dt
 +
$$
  
be a functional of the classical calculus of variations (cf. [[Variational calculus|Variational calculus]]), and let the [[Extremal|extremal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097460/w0974602.png" /> be continuously differentiable in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097460/w0974603.png" /> except at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097460/w0974604.png" /> itself, at which it has a corner. In this situation, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097460/w0974605.png" /> to be at least a weak local extremum for the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097460/w0974606.png" />, it is necessary that the equations
+
be a functional of the classical calculus of variations (cf. [[Variational calculus|Variational calculus]]), and let the [[Extremal|extremal]] $  x _ {0} ( t) $
 +
be continuously differentiable in a neighbourhood of the point $  \tau $
 +
except at the point $  \tau $
 +
itself, at which it has a corner. In this situation, for $  x _ {0} ( t) $
 +
to be at least a weak local extremum for the functional $  J( x) $,  
 +
it is necessary that the equations
  
<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/w097460/w0974607.png" /></td> </tr></table>
+
$$
 +
p( \tau - 0)  = p ( \tau + 0),
 +
$$
  
<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/w097460/w0974608.png" /></td> </tr></table>
+
$$
 +
H( \tau - 0)  = H ( \tau + 0),
 +
$$
  
 
where
 
where
  
<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/w097460/w0974609.png" /></td> </tr></table>
+
$$
 +
p( t)  = \
 +
 
 +
\frac{\partial  L( t , x _ {0} ( t), \dot{x} _ {0} ( t)) }{\partial  \dot{x} }
 +
 
 +
$$
  
 
and
 
and
  
<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/w097460/w09746010.png" /></td> </tr></table>
+
$$
 +
H( t)  = \
 +
( \dot{x} _ {0} ( t), p( t))- L ( t, x _ {0} ( t), \dot{x} _ {0} ( t)),
 +
$$
  
be satisfied at the corner point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097460/w09746011.png" />. These equations are known as the corner conditions of K. Weierstrass (1865) and G. Erdmann (1877) [[#References|[1]]].
+
be satisfied at the corner point $  \tau $.  
 +
These equations are known as the corner conditions of K. Weierstrass (1865) and G. Erdmann (1877) [[#References|[1]]].
  
 
The meaning of the Weierstrass–Erdmann corner conditions is that the canonical variables and the Hamiltonian are continuous at a corner point of the extremal; their meaning in classical mechanics is the continuity of momentum and of energy at a corner point.
 
The meaning of the Weierstrass–Erdmann corner conditions is that the canonical variables and the Hamiltonian are continuous at a corner point of the extremal; their meaning in classical mechanics is the continuity of momentum and of energy at a corner point.
  
In regular problems, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097460/w09746012.png" /> is a strictly convex function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097460/w09746013.png" />, the extremals cannot have corner points. Corner points appear if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097460/w09746014.png" /> and consequently the [[Weierstrass E-function|Weierstrass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097460/w09746015.png" />-function]], contains segments of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097460/w09746016.png" />. For the [[Lagrange problem|Lagrange problem]] with conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097460/w09746017.png" /> and [[Lagrange multipliers|Lagrange multipliers]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097460/w09746018.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097460/w09746019.png" /> in the Weierstrass–Erdmann corner conditions is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097460/w09746020.png" />.
+
In regular problems, when $  L $
 +
is a strictly convex function of $  \dot{x} $,  
 +
the extremals cannot have corner points. Corner points appear if $  L( t, x, \dot{x} ) $
 +
and consequently the [[Weierstrass E-function|Weierstrass $  {\mathcal E} $-
 +
function]], contains segments of $  \dot{x} $.  
 +
For the [[Lagrange problem|Lagrange problem]] with conditions $  \phi _ {i} ( t, x, \dot{x} ) = 0 $
 +
and [[Lagrange multipliers|Lagrange multipliers]] $  \lambda _ {i} ( t) $,  
 +
the $  L $
 +
in the Weierstrass–Erdmann corner conditions is replaced by $  \widetilde{L}  = L+ \sum _ {i} \lambda _ {i} \phi _ {i} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Erdmann,  "Ueber die unstetige Lösungen in der Variationsrechnung"  ''J. Reine Angew. Math.'' , '''82'''  (1877)  pp. 21–30</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O. Bolza,  "Lectures on the calculus of variations" , Chelsea, reprint  (1960)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.I. Akhiezer,  "The calculus of variations" , Blaisdell  (1962)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Erdmann,  "Ueber die unstetige Lösungen in der Variationsrechnung"  ''J. Reine Angew. Math.'' , '''82'''  (1877)  pp. 21–30</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O. Bolza,  "Lectures on the calculus of variations" , Chelsea, reprint  (1960)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.I. Akhiezer,  "The calculus of variations" , Blaisdell  (1962)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:28, 6 June 2020


Necessary conditions for an extremum, additional to the Euler equation, specified at points at which the extremal has a corner. Let

$$ J ( x) = \int\limits L ( t, x, \dot{x} ) dt $$

be a functional of the classical calculus of variations (cf. Variational calculus), and let the extremal $ x _ {0} ( t) $ be continuously differentiable in a neighbourhood of the point $ \tau $ except at the point $ \tau $ itself, at which it has a corner. In this situation, for $ x _ {0} ( t) $ to be at least a weak local extremum for the functional $ J( x) $, it is necessary that the equations

$$ p( \tau - 0) = p ( \tau + 0), $$

$$ H( \tau - 0) = H ( \tau + 0), $$

where

$$ p( t) = \ \frac{\partial L( t , x _ {0} ( t), \dot{x} _ {0} ( t)) }{\partial \dot{x} } $$

and

$$ H( t) = \ ( \dot{x} _ {0} ( t), p( t))- L ( t, x _ {0} ( t), \dot{x} _ {0} ( t)), $$

be satisfied at the corner point $ \tau $. These equations are known as the corner conditions of K. Weierstrass (1865) and G. Erdmann (1877) [1].

The meaning of the Weierstrass–Erdmann corner conditions is that the canonical variables and the Hamiltonian are continuous at a corner point of the extremal; their meaning in classical mechanics is the continuity of momentum and of energy at a corner point.

In regular problems, when $ L $ is a strictly convex function of $ \dot{x} $, the extremals cannot have corner points. Corner points appear if $ L( t, x, \dot{x} ) $ and consequently the Weierstrass $ {\mathcal E} $- function, contains segments of $ \dot{x} $. For the Lagrange problem with conditions $ \phi _ {i} ( t, x, \dot{x} ) = 0 $ and Lagrange multipliers $ \lambda _ {i} ( t) $, the $ L $ in the Weierstrass–Erdmann corner conditions is replaced by $ \widetilde{L} = L+ \sum _ {i} \lambda _ {i} \phi _ {i} $.

References

[1] G. Erdmann, "Ueber die unstetige Lösungen in der Variationsrechnung" J. Reine Angew. Math. , 82 (1877) pp. 21–30
[2] O. Bolza, "Lectures on the calculus of variations" , Chelsea, reprint (1960) (Translated from German)
[3] N.I. Akhiezer, "The calculus of variations" , Blaisdell (1962) (Translated from Russian)

Comments

See also Weierstrass conditions (for a variational extremum).

References

[a1] L. Cesari, "Optimization - Theory and applications" , Springer (1983)
[a2] G.M. Ewing, "Calculus of variations with applications" , Dover, reprint (1985)
[a3] Yu.P. Petrov, "Variational methods in optimum control theory" , Acad. Press (1968) pp. Chapt. IV (Translated from Russian)
How to Cite This Entry:
Weierstrass-Erdmann corner conditions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass-Erdmann_corner_conditions&oldid=16105
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article