View source for Weierstrass-Erdmann corner conditions

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>

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

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where

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and

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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]]].

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" />.

====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>



====Comments====
See also [[Weierstrass conditions (for a variational extremum)|Weierstrass conditions (for a variational extremum)]].

====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">  Yu.P. Petrov,   "Variational methods in optimum control theory" , Acad. Press  (1968)  pp. Chapt. IV  (Translated from Russian)</TD></TR></table>