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An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l0572701.png" />-dimensional differentiable submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l0572702.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l0572703.png" />-dimensional [[Symplectic manifold|symplectic manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l0572704.png" /> such that the exterior form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l0572705.png" /> specifying the symplectic structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l0572706.png" /> vanishes identically on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l0572707.png" /> (that is, for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l0572708.png" /> and any vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l0572709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l05727010.png" /> which are tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l05727011.png" /> at this point one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l05727012.png" />). In the most important case, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l05727013.png" /> with coordinates (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l05727014.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l05727015.png" />, the condition that the submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l05727016.png" /> given by the parametric equations
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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/l/l057/l057270/l05727017.png" /></td> </tr></table>
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is Lagrangian, has the form
+
An  $  n $-
 +
dimensional differentiable submanifold  $  L  ^ {n} $
 +
of a  $  2n $-
 +
dimensional [[Symplectic manifold|symplectic manifold]]  $  M  ^ {2n} $
 +
such that the exterior form  $  \omega $
 +
specifying the symplectic structure on  $  M  ^ {2n} $
 +
vanishes identically on  $  L  ^ {n} $(
 +
that is, for any point  $  x \in L  ^ {n} $
 +
and any vectors  $  X $
 +
and  $  Y $
 +
which are tangent to  $  L  ^ {n} $
 +
at this point one has $  \omega ( X , Y ) = 0 $).
 +
In the most important case, when  $  M  ^ {2n} = \mathbf R  ^ {2n} $
 +
with coordinates ( $  p _ {1} \dots p _ {n} , q _ {1} \dots q _ {n} $)
 +
and  $  \omega = \sum_{i=1}  ^ {n} dp _ {i} \wedge dq _ {i} $,
 +
the condition that the submanifold  $  L  ^ {n} $
 +
given by the parametric 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/l/l057/l057270/l05727018.png" /></td> </tr></table>
+
$$
 +
p _ {i}  = p _ {i} ( u _ {1} \dots u _ {n} ) ,\  q _ {i}  = q _ {i} ( u _ {1} \dots u _ {n} )
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057270/l05727019.png" /> is the [[Lagrange bracket|Lagrange bracket]].
+
is Lagrangian, has the form
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.P. Maslov,  "Théorie des perturbations et méthodes asymptotiques" , Dunod  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.P. Maslov,  "Introduction to the method of phase integrals (the WKB method)" , Moscow  (1965)  (In Russian)  (Appendix to the Russian translation of: J. Heading, The WKB method in the multi-dimensional case)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.P. Maslov,  M.V. Fedoryuk,  "Quasi-classical approximation for the equations of quantum mechanics" , Reidel  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.S. Mishchenko,  B.Yu. Sternin,  V.E. Shatalov,  "Lagrangian manifolds and the method of the canonical operator" , Moscow  (1978)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.I. [V.I. Arnol'd] Arnold,  A.B. [A.B. Givent'al] Giventhal,  "Symplectic geometry" , ''Dynamical Systems'' , '''IV''' , Springer  (1988)  (Translated from Russian)</TD></TR></table>
 
  
 +
$$
 +
[ u _ {i} , u _ {j} ]  =  0 ,\  i , j = 1 \dots n ,
 +
$$
  
 +
where  $  [ u _ {i} , u _ {j} ] $
 +
is the [[Lagrange bracket]].
  
 
====Comments====
 
====Comments====
Line 18: Line 46:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Libermann,  C.-M. Marle,  "Symplectic geometry and analytical mechanics" , Reidel  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I. Vaismann,  "Symplectic geometry and secondary characteristic classes" , Birkhäuser  (1987)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  V.P. Maslov, "Théorie des perturbations et méthodes asymptotiques" , Dunod  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.P. Maslov,  "Introduction to the method of phase integrals (the WKB method)" , Moscow  (1965)  (In Russian)  (Appendix to the Russian translation of: J. Heading, The WKB method in the multi-dimensional case)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.P. Maslov,  M.V. Fedoryuk,  "Quasi-classical approximation for the equations of quantum mechanics" , Reidel  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.S. Mishchenko,  B.Yu. Sternin,  V.E. Shatalov,  "Lagrangian manifolds and the method of the canonical operator" , Moscow  (1978)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.I. [V.I. Arnol'd] Arnold,  A.B. [A.B. Givent'al] Giventhal,  "Symplectic geometry" , ''Dynamical Systems'' , '''IV''' , Springer  (1988)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Libermann,  C.-M. Marle,  "Symplectic geometry and analytical mechanics" , Reidel  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I. Vaismann,  "Symplectic geometry and secondary characteristic classes" , Birkhäuser  (1987)</TD></TR>
 +
</table>

Latest revision as of 18:57, 9 January 2024


An $ n $- dimensional differentiable submanifold $ L ^ {n} $ of a $ 2n $- dimensional symplectic manifold $ M ^ {2n} $ such that the exterior form $ \omega $ specifying the symplectic structure on $ M ^ {2n} $ vanishes identically on $ L ^ {n} $( that is, for any point $ x \in L ^ {n} $ and any vectors $ X $ and $ Y $ which are tangent to $ L ^ {n} $ at this point one has $ \omega ( X , Y ) = 0 $). In the most important case, when $ M ^ {2n} = \mathbf R ^ {2n} $ with coordinates ( $ p _ {1} \dots p _ {n} , q _ {1} \dots q _ {n} $) and $ \omega = \sum_{i=1} ^ {n} dp _ {i} \wedge dq _ {i} $, the condition that the submanifold $ L ^ {n} $ given by the parametric equations

$$ p _ {i} = p _ {i} ( u _ {1} \dots u _ {n} ) ,\ q _ {i} = q _ {i} ( u _ {1} \dots u _ {n} ) $$

is Lagrangian, has the form

$$ [ u _ {i} , u _ {j} ] = 0 ,\ i , j = 1 \dots n , $$

where $ [ u _ {i} , u _ {j} ] $ is the Lagrange bracket.

Comments

Also often called Lagrangian submanifold.

References

[1] V.P. Maslov, "Théorie des perturbations et méthodes asymptotiques" , Dunod (1972) (Translated from Russian)
[2] V.P. Maslov, "Introduction to the method of phase integrals (the WKB method)" , Moscow (1965) (In Russian) (Appendix to the Russian translation of: J. Heading, The WKB method in the multi-dimensional case)
[3] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[4] V.P. Maslov, M.V. Fedoryuk, "Quasi-classical approximation for the equations of quantum mechanics" , Reidel (1981) (Translated from Russian)
[5] A.S. Mishchenko, B.Yu. Sternin, V.E. Shatalov, "Lagrangian manifolds and the method of the canonical operator" , Moscow (1978) (In Russian)
[6] V.I. [V.I. Arnol'd] Arnold, A.B. [A.B. Givent'al] Giventhal, "Symplectic geometry" , Dynamical Systems , IV , Springer (1988) (Translated from Russian)
[a1] P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) (Translated from French)
[a2] I. Vaismann, "Symplectic geometry and secondary characteristic classes" , Birkhäuser (1987)
How to Cite This Entry:
Lagrangian manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lagrangian_manifold&oldid=16390
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article