TITLE: What is a Lagrangian submanifold intuitively?
QUESTION [68 upvotes]: What are good ways to think about Lagrangian submanifolds?
Why should one care about them? 
More generally: same questions about (co)isotropic ones. 
Answers from a classical mechanics point of view would be especially welcome.

REPLY [3 votes]: As mentioned in previous answers - Lagrangian submanifolds encode vast amount of information on the symplectic geometry of the ambient symplectic manifold $M$ they live in (like sheaves on an algebraic manifold) - so studying them is really an essential feature of symplectic geometry. 
As of themselves, Lagrangian submanifolds also admit, for instance, some special intersection properties - which are typically proved by pseudo-holomorphic techniques or the more "modern machinery" of Floer homology. If you are interested in the subject a great place to start is Biran's Lagrangian non-intersections (2005).
But, even if your interest lies in algebraic geometry, in general, Lagrangian submanifolds can still prove very interesting to you - due to the homological mirror symmetry conjectures of Kontsevich - which suggest that sheaves (or more generally, elements of $\mathcal{D}^b(X)$) on one manifold W (e.g Calabi-Yau) should literally 
translate to Lagrangian submanifolds (or elements of Fukaya category) in another manifold $M$ - via a mirror functor. This led to some fascinating developments in both symplectic and algebraic geometry.