a Lagrangian submanifold
/ luh-GRAHN-jee-un /
On a symplectic manifold, the 2-form omega measures area, and it pairs vectors antisymmetrically. A Lagrangian submanifold is a submanifold of exactly half the dimension on which this area-measurement completely collapses — restrict omega to the submanifold and you get zero. These are the largest such 'omega-blind' subspaces possible, and they are everywhere in mechanics: graphs of generating functions, level sets of conserved quantities, the zero section of a cotangent bundle, and the configurations that solutions to mechanical problems trace out.
Precisely, in a symplectic manifold (M^{2n}, omega), a submanifold L is isotropic if omega restricted to L vanishes (omega(v, w) = 0 for all v, w tangent to L), which forces dim L <= n. It is Lagrangian if it is isotropic of the maximal dimension n, i.e. dim L = (1/2) dim M and omega|_L = 0. Equivalently, at each point the tangent space T_p L equals its own symplectic orthogonal complement (T_p L)^omega. Standard examples: in (R^{2n}, sum dp_i ^ dq_i), the q-plane {p = 0} and the p-plane {q = 0} are both Lagrangian; in a cotangent bundle T*N, the zero section and the graph of any closed 1-form are Lagrangian, and a fiber T_p^* N is Lagrangian too.
Why they are the central objects: Weinstein's creed is that 'everything is a Lagrangian submanifold' — a symplectomorphism f: M -> M, for instance, is encoded as the Lagrangian graph {(x, f(x))} inside the product M x M with the twisted symplectic form. Lagrangian intersection theory (how many points two Lagrangians must share) is the geometric content of Floer homology and the Arnold conjecture. A frequent error is to think any half-dimensional submanifold qualifies: it does not — the vanishing omega|_L = 0 is a strong constraint, and a generic half-dimensional submanifold is not isotropic, let alone Lagrangian.
In the cotangent bundle T*N of any manifold N, the canonical symplectic form is omega = -d lambda where lambda is the tautological 1-form. The graph of a 1-form alpha on N, viewed as a submanifold of T*N, is Lagrangian if and only if d alpha = 0, i.e. alpha is closed. In particular the zero section (alpha = 0) is always Lagrangian, and the graph of dh for any function h is Lagrangian — these graphs are the 'generating functions' that encode canonical transformations.
In T*N a closed 1-form's graph is Lagrangian; the zero section and fibers are too.
Isotropic, Lagrangian, and coisotropic are three distinct dimension regimes: isotropic has dim <= n with omega|_L = 0; coisotropic has the orthogonal (T_p L)^omega contained in T_p L (dim >= n); Lagrangian is the exact balance dim = n where both hold. Lagrangian is the borderline case, not a generic submanifold.