TITLE: When do you go hunting for Lagrangian submanifolds?
QUESTION [12 upvotes]: Similar to this question, I'm trying to figure out why one would be interested in Lagrangian submanifolds. But from a more geometric point of view. My best find so far is Exercise 12.4 in McDuff, Salamon, which is to

... construct a symplectic embedding $$B^{2n+2}(r)\hookrightarrow \mathbb{T}(1)\times\mathbb{R}^{2n}$$ in the following way. Find a linear Lagrangian subspace $L$ of $\mathbb{R}^2\times \mathbb{R}^{2n}$ whose $\epsilon$-neighbourhood $L_\epsilon$ projections injectively into $\mathbb{R}^2/\mathbb{Z}^2\times \mathbb{R}^{2n}=\mathbb{T}^2(1)\times \mathbb{R}^{2n}$. Then consider a composite mapping $$B^{2n+2}(r)\hookrightarrow L_\epsilon\hookrightarrow \mathbb{T}^2(1)\times\mathbb{R}^{2n}.$$

A bit earlier they explained that one may embed $B^{2n}(1)\hookrightarrow B^n(\delta)\times \mathbb{R}^n$ by $(x,y)\mapsto (\delta x,\delta^{-1}y)$.
I suspect that the reason people care about Lagrangian submanifolds is that there are more situations where one may move a construction from one place to another by finding appropriate Lagrangian submanifolds.
So my question is: What are other geometric problems in which the existence of a certain Lagrangian manifold is used in a manner similar to the above? That is, when do you go hunting for Lagrangian submanifolds?
Edit: To clarify, I'm not that much interested in generalities, but in concrete geometrical problems where a Lagrangian does the trick.

REPLY [4 votes]: Another place where Lagrangian submanifolds arise naturally is in cotangent bundles. If $M\subset N$ is a submanifold then you can define the conormal bundle of $M$ to be the set of covectors which annihilate $TM$. This is Lagrangian (in fact an exact Lagrangian). You also get a Legendrian submanifold when you intersect this with the unit cosphere bundle (Legendrians are the contact-geometric cousins of Lagrangian submanifolds). Legendrians and Lagrangians are very intimately related and you might as well go hunting for both.
Now contact geometric invariants of this Legendrian (or symplectic invariants of the Lagrangian) conormal bundle are smooth invariants of the embedded submanifold $M$. In particular, Ng's knot contact homology (when $M\subset N$ is a knot in a 3-manifold) gives extremely interesting knot invariants related (see recent work of Aganagic-Ekholm-Ng-Vafa based on ideas of Witten) to Chern-Simons theory.
Just as in Jean-Claude Sikorav's answer (though in a completely orthogonal direction) this indicates that (exact) Lagrangian submanifolds of cotangent bundles remember much about smooth topology in the base.