Theorem of Weinstein

In the symplectic geometry , one of the Hamiltonian mechanics developed branch of mathematics , which is set by Weinstein (ger .: Weinstein's neighborhood theorem ) is a theorem , the environment Lagrangian submanifolds in symplectic manifolds describes.

Theorem of Weinstein

Let be a symplectic manifold and a Lagrangian submanifold . ${\displaystyle (M,\omega )}$ ${\displaystyle L\subset M}$

Then there is a neighborhood of that is symplectomorphic to a neighborhood of the zero intersection in . ${\displaystyle U(L)}$ ${\displaystyle L}$ ${\displaystyle T^{*}L}$

The cotangent bundle is provided with the canonical symplectic form in the canonical coordinates . ${\displaystyle T^{*}L}$ ${\displaystyle \sum _{i}dq_{i}\wedge dp_{i}}$ ${\displaystyle q,p}$

literature

• Alan Weinstein : Symplectic manifolds and their Lagrangian submanifolds. Advances in Math. 6, 329-346 (1971).
• Ana Cannas da Silva : Introduction to symplectic and Hamiltonian geometry. Publicações Matemáticas do IMPA. Rio de Janeiro, 2003. ISBN 85-244-0195-8