Lagrangian manifold

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)