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A collection of solutions of the [[Euler equation|Euler equation]], depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e0372301.png" /> arbitrary constants and filling without mutual intersections some part of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e0372302.png" />-dimensional space. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e0372303.png" /> is the number of unknown functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e0372304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e0372305.png" />, on which the functional to be minimized,
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A collection of solutions of the [[Euler equation|Euler equation]], depending on $n$ arbitrary constants and filling without mutual intersections some part of the $(n+1)$-dimensional space. Here $n$ is the number of unknown functions $y_i(x)$, $i=1,\dots,n$, on which the functional to be minimized,
  
<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/e/e037/e037230/e0372306.png" /></td> </tr></table>
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$$J(y_1,\dots,y_n)=\int\limits_{x_1}^{x_2}F(x,y_1,\dots,y_n,y'_1,\dots,y'_n)dx,$$
  
depends. Euler's equation is understood in the vector sense, that is, it is a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e0372307.png" /> ordinary differential equations of the second order:
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depends. Euler's equation is understood in the vector sense, that is, it is a system of $n$ ordinary differential equations of the second order:
  
<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/e/e037/e037230/e0372308.png" /></td> </tr></table>
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$$F_{y_i}-\frac d{dx}F_{y'_t}=0,\quad i=1,\dots,n.$$
  
 
Two methods for constructing an extremal set are indicated below.
 
Two methods for constructing an extremal set are indicated below.
  
Let the object of investigation be a pencil of extremals emanating from a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e0372309.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e03723010.png" />-dimensional space. If the extremals of the pencil do not intersect each other in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e03723011.png" /> (except at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e03723012.png" />), then they form an extremal set (a central extremal set) in this neighbourhood.
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Let the object of investigation be a pencil of extremals emanating from a given point $M_0(x_0,y_0)$ in an $(n+1)$-dimensional space. If the extremals of the pencil do not intersect each other in some neighbourhood of $M_0$ (except at $M_0$), then they form an extremal set (a central extremal set) in this neighbourhood.
  
Another method of constructing extremals consists in constructing the set of extremals that are transversal to a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e03723013.png" /> given in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e03723014.png" />-dimensional space by an equation
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Another method of constructing extremals consists in constructing the set of extremals that are transversal to a surface $S_0$ given in the $(n+1)$-dimensional space by an 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/e/e037/e037230/e03723015.png" /></td> </tr></table>
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$$\phi(x,y)=0.$$
  
 
If at every point of this surface the transversality conditions
 
If at every point of this surface the transversality conditions
  
<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/e/e037/e037230/e03723016.png" /></td> </tr></table>
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$$\frac{F-\sum_{i=1}^ny'_iF_{y'_i}}{\phi_x}=\frac{F_{y'_1}}{\phi_{y_1}}=\dots=\frac{F_{y'_n}}{\phi_{y_n}},$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e03723017.png" /> in total, determine the value of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e03723018.png" /> derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e03723019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e03723020.png" />, then by taking these values as initial values of the derivatives one can draw through a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e03723021.png" /> an extremal that intersects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037230/e03723022.png" /> transversally. If in a neighbourhood of this surface the above extremals do not intersect each other, then they form an extremal set (an ordinary, or proper extremal set).
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$n$ in total, determine the value of the $n$ derivatives $y'_i$, $i=1,\dots,n$, then by taking these values as initial values of the derivatives one can draw through a point of $S_0$ an extremal that intersects $S_0$ transversally. If in a neighbourhood of this surface the above extremals do not intersect each other, then they form an extremal set (an ordinary, or proper extremal set).
  
 
The construction of an extremal set is the starting point in the discussion of questions connected with the construction of a field of extremals (cf. [[Extremal|Extremal]]). An extremal set is an [[Extremal field|extremal field]] if there is a one-parameter family of surfaces that intersect the extremals of the set transversally.
 
The construction of an extremal set is the starting point in the discussion of questions connected with the construction of a field of extremals (cf. [[Extremal|Extremal]]). An extremal set is an [[Extremal field|extremal field]] if there is a one-parameter family of surfaces that intersect the extremals of the set transversally.

Latest revision as of 16:38, 24 November 2018

A collection of solutions of the Euler equation, depending on $n$ arbitrary constants and filling without mutual intersections some part of the $(n+1)$-dimensional space. Here $n$ is the number of unknown functions $y_i(x)$, $i=1,\dots,n$, on which the functional to be minimized,

$$J(y_1,\dots,y_n)=\int\limits_{x_1}^{x_2}F(x,y_1,\dots,y_n,y'_1,\dots,y'_n)dx,$$

depends. Euler's equation is understood in the vector sense, that is, it is a system of $n$ ordinary differential equations of the second order:

$$F_{y_i}-\frac d{dx}F_{y'_t}=0,\quad i=1,\dots,n.$$

Two methods for constructing an extremal set are indicated below.

Let the object of investigation be a pencil of extremals emanating from a given point $M_0(x_0,y_0)$ in an $(n+1)$-dimensional space. If the extremals of the pencil do not intersect each other in some neighbourhood of $M_0$ (except at $M_0$), then they form an extremal set (a central extremal set) in this neighbourhood.

Another method of constructing extremals consists in constructing the set of extremals that are transversal to a surface $S_0$ given in the $(n+1)$-dimensional space by an equation

$$\phi(x,y)=0.$$

If at every point of this surface the transversality conditions

$$\frac{F-\sum_{i=1}^ny'_iF_{y'_i}}{\phi_x}=\frac{F_{y'_1}}{\phi_{y_1}}=\dots=\frac{F_{y'_n}}{\phi_{y_n}},$$

$n$ in total, determine the value of the $n$ derivatives $y'_i$, $i=1,\dots,n$, then by taking these values as initial values of the derivatives one can draw through a point of $S_0$ an extremal that intersects $S_0$ transversally. If in a neighbourhood of this surface the above extremals do not intersect each other, then they form an extremal set (an ordinary, or proper extremal set).

The construction of an extremal set is the starting point in the discussion of questions connected with the construction of a field of extremals (cf. Extremal). An extremal set is an extremal field if there is a one-parameter family of surfaces that intersect the extremals of the set transversally.

References

[1] V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)


Comments

References

[a1] I.M. Gel'fand, S.V. Fomin, "Calculus of variations" , Prentice-Hall (1963) (Translated from Russian)
How to Cite This Entry:
Extremal set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extremal_set&oldid=15984
This article was adapted from an original article by I.B. Vapnyarskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article