TITLE: Why are Lagrangian submanifolds called Lagrangian?
QUESTION [24 upvotes]: Much of the terminology in symplectic geometry comes from classical mechanics: the symplectic manifold is modeled on a cotangent bundle $T^*N$ of some configuration space $N$ with local position coordinates $q_i$ and momentum coordinates $p_i$ such that $\omega = \sum_i dq_i \wedge dp_i$. Then, a Hamiltonian - a.k.a. a smooth "energy" function - on this phase space induces a flow satisfying Hamilton's equations from classical mechanics: 
$$\frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i} \hskip{1 in} \frac{dq_i}{dt} = \frac{\partial H}{\partial p_i}$$ 
In this physical interpretation, what are Lagrangian submanifolds? In particular, why are they named Lagrangian? Is there a relationship between this notion and the Lagrangian formulation of mechanics? 
(Note: this question has a lot of great answers providing some physical or geoemtric intuition for Lagrangian submanifolds - in a cotangent bundle, both fibers and the images of closed sections (closed $1$-forms on $N$) are motivating examples of Lagrangian submanifolds - but it does not address the source of the terminology.)

REPLY [11 votes]: This echos the 2017 comments, but since the question has now been bumped to the front page it might be helpful to give the actual source in Maslov's book [1].   
[1] V.P. Maslov, Perturbation Theory and Asymptotic Methods (in Russian, Moscow, 1965); Théorie des pertubations et méthodes asymptotique (French translation, Paris, 1972).

The Lagrangian manifold (variété lagrangienne) is introduced on page 114-115: