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A transformation of the equations of plane motion of a material point under the action of a force that varies with the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012810/a0128101.png" /> of the point only. The Appell transformation is a homographic transformation of the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012810/a0128102.png" /> into the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012810/a0128103.png" />:
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<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/a/a012/a012810/a0128104.png" /></td> </tr></table>
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which involves the substitution of the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012810/a0128105.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012810/a0128106.png" /> in accordance with the formula
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A transformation of the equations of plane motion of a material point under the action of a force that varies with the coordinates  $  x, y $
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of the point only. The Appell transformation is a homographic transformation of the coordinates  $  x, y $
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into the coordinates  $  x _ {1} , y _ {1} $:
  
<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/a/a012/a012810/a0128107.png" /></td> </tr></table>
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$$
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x _ {1}  =
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\frac{ax + by + c }{a  ^ {\prime\prime} x + b  ^ {\prime\prime} y + c  ^ {\prime\prime} }
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,\ \
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y _ {1}  =
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\frac{a  ^  \prime  x + b  ^  \prime  y + c  ^  \prime  }{a ^ {\prime\prime} x + b  ^ {\prime\prime} y + c  ^ {\prime\prime} }
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,
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$$
  
As a result of the Appell transformation, the motion of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012810/a0128108.png" />, acted upon by a force <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012810/a0128109.png" /> that depends only on the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012810/a01281010.png" />, corresponds to the motion of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012810/a01281011.png" /> acted upon by the force <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012810/a01281012.png" />. A transformation of the equations of motion of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012810/a01281013.png" /> with the aid of formulas of general form
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which involves the substitution of the time  $  t _ {1} $
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for  $  t $
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in accordance with the formula
  
<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/a/a012/a012810/a01281014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$
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k  dt _ {1}  =
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\frac{dt}{( a ^ {\prime\prime} x + b  ^ {\prime\prime} y +c  ^ {\prime\prime} )  ^ {2} }
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.
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$$
  
yields the equations of motion of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012810/a01281015.png" /> under the effect of a force that depends only on the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012810/a01281016.png" /> only if (*) is a homographic transformation. The Appell transformation is used to solve Bertrand's problem on the determination of the forces under which the trajectories of motion of material points become conic sections. The Appell transformation is named after P.E. Appell.
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As a result of the Appell transformation, the motion of a point  $  ( x _ {1} , y _ {1} ) $,
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acted upon by a force  $  F _ {1} ( x _ {1} , y _ {1} ) $
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that depends only on the coordinates  $  x _ {1} , y _ {1} $,
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corresponds to the motion of the point  $  (x, y) $
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acted upon by the force  $  F(x, y) $.
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A transformation of the equations of motion of a point  $  (x, y) $
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with the aid of formulas of general form
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$$ \tag{* }
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x _ {1}  = \phi ( x , y ),\  y _ {1}  = \psi ( x , y ),\ \
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dt _ {1}  = \lambda ( x , y )  dt ,
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$$
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yields the equations of motion of the point  $  ( x _ {1} , y _ {1} ) $
 +
under the effect of a force that depends only on the coordinates $  x _ {1} , y _ {1} $
 +
only if (*) is a homographic transformation. The Appell transformation is used to solve Bertrand's problem on the determination of the forces under which the trajectories of motion of material points become conic sections. The Appell transformation is named after P.E. Appell.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.E. Appell,  "De l'homographie en mécanique"  ''Amer. J. Math.'' , '''12'''  (1890)  pp. 103–114</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.E. Appell,  "Sur les lois de forces centrales faisant décrire à leur point d'application une conique quelles que soient les conditions initiales"  ''Amer. J. Math.'' , '''13'''  (1891)  pp. 153–158</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.E. Appell,  "De l'homographie en mécanique"  ''Amer. J. Math.'' , '''12'''  (1890)  pp. 103–114</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.E. Appell,  "Sur les lois de forces centrales faisant décrire à leur point d'application une conique quelles que soient les conditions initiales"  ''Amer. J. Math.'' , '''13'''  (1891)  pp. 153–158</TD></TR></table>

Revision as of 18:47, 5 April 2020


A transformation of the equations of plane motion of a material point under the action of a force that varies with the coordinates $ x, y $ of the point only. The Appell transformation is a homographic transformation of the coordinates $ x, y $ into the coordinates $ x _ {1} , y _ {1} $:

$$ x _ {1} = \frac{ax + by + c }{a ^ {\prime\prime} x + b ^ {\prime\prime} y + c ^ {\prime\prime} } ,\ \ y _ {1} = \frac{a ^ \prime x + b ^ \prime y + c ^ \prime }{a ^ {\prime\prime} x + b ^ {\prime\prime} y + c ^ {\prime\prime} } , $$

which involves the substitution of the time $ t _ {1} $ for $ t $ in accordance with the formula

$$ k dt _ {1} = \frac{dt}{( a ^ {\prime\prime} x + b ^ {\prime\prime} y +c ^ {\prime\prime} ) ^ {2} } . $$

As a result of the Appell transformation, the motion of a point $ ( x _ {1} , y _ {1} ) $, acted upon by a force $ F _ {1} ( x _ {1} , y _ {1} ) $ that depends only on the coordinates $ x _ {1} , y _ {1} $, corresponds to the motion of the point $ (x, y) $ acted upon by the force $ F(x, y) $. A transformation of the equations of motion of a point $ (x, y) $ with the aid of formulas of general form

$$ \tag{* } x _ {1} = \phi ( x , y ),\ y _ {1} = \psi ( x , y ),\ \ dt _ {1} = \lambda ( x , y ) dt , $$

yields the equations of motion of the point $ ( x _ {1} , y _ {1} ) $ under the effect of a force that depends only on the coordinates $ x _ {1} , y _ {1} $ only if (*) is a homographic transformation. The Appell transformation is used to solve Bertrand's problem on the determination of the forces under which the trajectories of motion of material points become conic sections. The Appell transformation is named after P.E. Appell.

References

[1] P.E. Appell, "De l'homographie en mécanique" Amer. J. Math. , 12 (1890) pp. 103–114
[2] P.E. Appell, "Sur les lois de forces centrales faisant décrire à leur point d'application une conique quelles que soient les conditions initiales" Amer. J. Math. , 13 (1891) pp. 153–158
How to Cite This Entry:
Appell transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Appell_transformation&oldid=12622
This article was adapted from an original article by L.N. Sretenskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article