What is a Lagrangian submanifold intuitively?

I want to answer the question "why should I care about Lagrangians" by showing how they arise when you try to answer a very natural question which in some sense lies at the heart of classical mechanics.

Suppose we consider some physical system in which the possible configurations correspond to points in a manifold $M$. Given two points $p, q \in M$ a natural question is to ask, is there some initial momentum one can select at $p$ under which after 1 second the configuration will have evolved into the configuration $q$?

In the Hamiltonian formulation of classical mechanics, the position and momentum of a system is described by the points of the cotangent bundle $T^*M$. This is a symplectic manifold in a natural way. The physics of the system are described by a so-called Hamiltonian function $H \colon T^*M \to \mathbb{R}$ and the evolution of the system is given by the flow of the Hamiltonian vector field $X_H$ associated to $H$. We can now rephrase our question as follows: is there a point $x$ in $T^*_pM$ such that the time-1 flow of $X_H$ starting at $x$ lies on the fibre $T^*_qM$? In other words, if I look at the image of $T^*_pM$ under the time-1 flow of $X_H$, does it intersect $T^*_qM$?

What does this have to do with Lagrangians? Well the fibres $T^*_pM$ and $T^*_qM$ are special cases of Lagrangian submanifolds. The image of $T^*_pM$ under the flow of $X_H$ is certainly no longer a fibre but, since the flow of $X_H$ preserves the symplectic form, the image of $T^*_pM$ remains a Lagrangian submanifold.

From this point of view you can think of a Lagrangian submanifold as a generalisation of "the set of possible initial momenta of a given point in configuration space". Moreover, this gerenalisation is forced on you, even if you only care about the very natural question described above. Finally, I hope this answer explains not only why Lagrangians are interesting, but why the possible intersections of different Lagrangians is interesting.

(Credit where credit is due: I guess this picture goes back at least to Arnold. I heard it in a seminar the other week given by Fukaya. The study of intersections of Lagrangians has become a huge industry recently, building essentially on the pioneering ideas of Floer.)

Everything is a lagrangian submanifold (A. Weinstein's lagrangian creed...) Indeed, every manifold is the lagrangian zero section of its cotangent bundle ;)

But more serious: Weinstein's tubular neighbourhood theorem states that every lagrangian submanifold in a symplectic manifold has a neighbourhood symplectomorphic to a neighbourhood of the zero section of the cotangent bundle.

Occurences of lagrangian submanifolds are indeed manifold: they arise as semiclassical support for certain FIO's and can also be thought of as semiclassical version of states in quantum mechanics via the WKB expansion. This point of view is exemplified a lot in the nice booklet of Bates and Weinstein.

Another occurence of coisotropics is in constraint mechanics: In Dirac's theory of constraints a coisotropic submanifold is what he calls a first class constraint. They arise very often in geometric mechanics with degenerate Lagrangians etc.

They are also the natural starting point for reduction: this is perhaps the more modern "coisotropic creed" of Lu (everything is a...)

Finally, to make the deformation quantization aspects a bit more precise: if you are looking for a submanifold $C \subseteq M$ in a Poisson manifold with star product $\star$ which allows for a (say) left module structure on $C^\infty(C)[[\hbar]]$ in such a way that the zeroth order of the module structure is the usual multiplication by the restriction, then you can show quite easily that $C$ has to be coisotropic. Martin Bordemann has a nice point of view how this relates to a theory of quantizing reduciton etc. in his (french!) big preprint :) In particular, the classical vanishing ideal becomes deformed into a left ideal for the star product (this was, I guess, essentially Lu's suggestion)

Note however, that there are other left ideals not of this form, e.g. the Gel'fand ideals of positive functionals, which can be much smaller.

The role of the lagrangian submanifolds $L$ in this context is that the corresponding representation on $C^\infty(L)[[\hbar]]$ becomes "irreducible" in a meaningful way (trivial commutant in the local operators). However, this only makes sense in the symplectic surrounding. In a truly Poisson manifold, only coisotropic makes sense.

Lagrangian submanifolds arise naturally in Hamiltonian Mechanics, because of the classical Arnold-Liouville theorem. Let me state it here:

Theorem (Arnold-Liouville). Let $(M, \omega, H)$ be an integrable system of dimension $2n$ with integrals of motion $f_1=H$, $f_2, \ldots, f_n$. Let $c \in \mathbb{R}^n$ be a regular value of $f:=(f_1, \ldots, f_n)$. Then the corresponding level $f^{-1}(c)$ is a Lagrangian submanifold of $M$.

Geometrically this means that, locally around the regular value $c$, the map $f \colon M \to \mathbb{R}^n$ collecting the integrals of motion is a Lagrangian fibration, i.e. it is locally trivial and the fibres are Lagrangian submanifolds.

Furthermore, one also shows that the connected components of $f^{-1}(c)$ are of the form $\mathbb{R}^{n-k} \times \mathbb{T}^k$, where $0 \leq k \leq n$ and $\mathbb{T}^k$ is a $k$-dimensional torus. In particular, every compact component must be a lagrangian torus.

For a proof of this result, see for instance the book by Ana Canas Da Silva "Lectures on symplectic geometry".