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I see references to Lagrangian Submanifolds in the literature but don't know what they are. Is there a relation to Lagrangian Tori (which I also don't know what are). Could someone give a definition that includes an intuitive explanation for what their use is?

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It's not entirely clear from your question whether you found these definitions: en.wikipedia.org/wiki/… and planetmath.org/encyclopedia/LagrangianSubmanifold.html . –  joriki Apr 11 '11 at 5:53

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It's a special kind of submanifold of a symplectic manifold; see the definition on Wikipedia. There was a question about this on Math Overflow not long ago, where you will find some good answers.

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Since again you posted this answer 2 minutes after I posted a comment, let me explain :-). At first I posted it as an answer (with our discussion about comments and answers in mind), but then I realized that the question asked for a lot more than what I wrote, and in fact could be interpreted such that it doesn't actually ask for a raw definition, so I decided to turn it into a comment instead. –  joriki Apr 11 '11 at 6:02
    
@joriki: Hello again! No need to apologize. Writing that explanation must have been more work for you than writing the comment in the first place. ;) –  Hans Lundmark Apr 11 '11 at 8:21

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