Symplectic manifolds with Lagrangian fibration (Q1084678)

Let (N,\(\omega)\) be a symplectic manifold and let \(\pi\) : \(N\to M\) be a surjective submersion. If each fibre of \(\pi\) is a Lagrangian submanifold of (N,\(\omega)\), then the quadruple (N,\(\pi\),M,\(\omega)\) is said to be a Lagrangian fibration. When \(N=T^*M\) and \(\pi\) is the canonical projection, we have the standard example of a Lagrangian fibration. A structure theorem for certain symplectic manifolds with a Lagrangian fibration is proved. As a corollary, a characterization of the cotangent bundles among symplectic manifolds with Lagrangian fibration is obtained. These results are then applied to two situations in classical mechanics: the Legendre transformation and Hamilton-Jacobi theory.