Lagrangian submanifold

Definition:
Let $(M,\omega )$ be a $2n$-dimensional symplectic manifold. A submanifold $L\subset M$ is called Lagrangian if:

1. $\dim (L)=n$.
2. ${\iota }^{\ast }\omega =0$, where $\iota :L\hookrightarrow M$ is the inclusion.
   (Equivalently, the symplectic form vanishes on any pair of vectors tangent to $L$).

Significance in mechanics:

• Liouville tori: The level sets of a completely integrable system (see integrable system#Liouville Foliation and Invariant Manifolds) are Lagrangian submanifolds.
• Configuration space: In the cotangent bundle $T^{\ast }Q$ with the canonical symplectic form $\omega =d\theta$, the zero section (representing just the configuration space $Q$) is a Lagrangian submanifold.

Relation to integrability:
The fact that $\{f_i,f_j\}=0$ (involution) is geometrically equivalent to the statement that the distribution spanned by the Hamiltonian vector fields $X_{f_i}$ is isotropic (the symplectic form vanish). Since there are $n$ independent functions, this distribution is maximal, making the integral leaves (the level sets) Lagrangian.