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.



What is a Lagrangian submanifold intuitively?
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".


