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Difference between revisions of "Poincaré last theorem"

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m (moved Poincare last theorem to Poincaré last theorem over redirect: accented title)
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p0730601.png" /> be an annulus in the plane bounded by circles with radii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p0730602.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p0730603.png" /> and let a mapping from this domain onto itself (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p0730604.png" /> is the polar angle), given by
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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/p/p073/p073060/p0730605.png" /></td> </tr></table>
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Let  $  K $
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be an annulus in the plane bounded by circles with radii  $  r = a $
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and  $  r = b $
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and let a mapping from this domain onto itself ( $  \theta $
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is the polar angle), given by
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$$
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\widetilde{r}  = \phi ( r , \theta ) ,\ \
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\widetilde \theta    = \psi ( r , \theta ) ,
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$$
  
 
satisfy the conditions: 1) the mapping preserves area; 2) each boundary circle maps onto itself,
 
satisfy the conditions: 1) the mapping preserves area; 2) each boundary circle maps onto itself,
  
<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/p/p073/p073060/p0730606.png" /></td> </tr></table>
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$$
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\phi ( a , \theta )  = a ,\ \
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\phi ( b , \theta )  = b ;
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$$
  
and 3) the points with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p0730607.png" /> move counter-clockwise and the points with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p0730608.png" /> clockwise, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p0730609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p07306010.png" />. Then this mapping has two fixed points. More generally, instead of preserving area one can require that no subdomain maps to a proper subset of itself.
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and 3) the points with $  r = a $
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move counter-clockwise and the points with $  r = b $
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clockwise, that is, $  \psi ( a , \theta ) > \theta $,  
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$  \psi ( b , \theta ) < \theta $.  
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Then this mapping has two fixed points. More generally, instead of preserving area one can require that no subdomain maps to a proper subset of itself.
  
 
This theorem was stated by H. Poincaré [[#References|[1]]] in 1912 in connection with certain problems of celestial mechanics; it was proved by him in a series of particular cases but he did not, however, obtain a general proof of this theorem. The paper was sent by Poincaré to an Italian journal (see [[#References|[1]]]) two weeks before his death, and the author expressed his conviction, in an accompanying letter to the editor, of the validity of the theorem in the general case.
 
This theorem was stated by H. Poincaré [[#References|[1]]] in 1912 in connection with certain problems of celestial mechanics; it was proved by him in a series of particular cases but he did not, however, obtain a general proof of this theorem. The paper was sent by Poincaré to an Italian journal (see [[#References|[1]]]) two weeks before his death, and the author expressed his conviction, in an accompanying letter to the editor, of the validity of the theorem in the general case.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  "Sur un théorème de géometrie"  ''Rend. Circ. Mat. Palermo'' , '''33'''  (1912)  pp. 375–407</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Birkhoff,  "Proof of Poincaré's geometric theorem"  ''Trans. Amer. Math. Soc.'' , '''14'''  (1913)  pp. 14–22</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.A. Pars,  "A treatise on analytical dynamics" , Heinemann , London  (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  "Sur un théorème de géometrie"  ''Rend. Circ. Mat. Palermo'' , '''33'''  (1912)  pp. 375–407</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Birkhoff,  "Proof of Poincaré's geometric theorem"  ''Trans. Amer. Math. Soc.'' , '''14'''  (1913)  pp. 14–22</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.A. Pars,  "A treatise on analytical dynamics" , Heinemann , London  (1965)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 08:06, 6 June 2020


Let $ K $ be an annulus in the plane bounded by circles with radii $ r = a $ and $ r = b $ and let a mapping from this domain onto itself ( $ \theta $ is the polar angle), given by

$$ \widetilde{r} = \phi ( r , \theta ) ,\ \ \widetilde \theta = \psi ( r , \theta ) , $$

satisfy the conditions: 1) the mapping preserves area; 2) each boundary circle maps onto itself,

$$ \phi ( a , \theta ) = a ,\ \ \phi ( b , \theta ) = b ; $$

and 3) the points with $ r = a $ move counter-clockwise and the points with $ r = b $ clockwise, that is, $ \psi ( a , \theta ) > \theta $, $ \psi ( b , \theta ) < \theta $. Then this mapping has two fixed points. More generally, instead of preserving area one can require that no subdomain maps to a proper subset of itself.

This theorem was stated by H. Poincaré [1] in 1912 in connection with certain problems of celestial mechanics; it was proved by him in a series of particular cases but he did not, however, obtain a general proof of this theorem. The paper was sent by Poincaré to an Italian journal (see [1]) two weeks before his death, and the author expressed his conviction, in an accompanying letter to the editor, of the validity of the theorem in the general case.

References

[1] H. Poincaré, "Sur un théorème de géometrie" Rend. Circ. Mat. Palermo , 33 (1912) pp. 375–407
[2] G. Birkhoff, "Proof of Poincaré's geometric theorem" Trans. Amer. Math. Soc. , 14 (1913) pp. 14–22
[3] L.A. Pars, "A treatise on analytical dynamics" , Heinemann , London (1965)

Comments

A proof of Poincaré's last theorem is in [2]. It is also known as the Poincaré–Birkhoff fixed-point theorem.

References

[a1] V.I. Arnol'd, A. Avez, "Problèmes ergodiques de la mécanique classique" , Gauthier-Villars (1967) pp. §20.5; Append. 29 (Translated from Russian)
[a2] G.D. Birkhoff, "Dynamical systems" , Amer. Math. Soc. (1927)
How to Cite This Entry:
Poincaré last theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_last_theorem&oldid=48205
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article