  Let $\pi : E\to M$ be a smooth fiber bundle whose total space is a symplectic
manifold and whose fibers are Lagrangian. Let $L$ be an embedded Lagrangian
submanifold of $E$. In the paper we address the following question: how can one
simplify the singularities of the projection $\pi: L \to M$ by a Hamiltonian
isotopy of $L$ inside $E$?
  We give an answer in the case when $dim L = 2$ and both $L$ and $M$ are
orientable. A weaker version of the result is proved in the higher-dimensional
case. Similar results hold in the contact category.
  As a corollary one gets an answer to one of the questions of V.Arnold about
the four cusps on the caustic in the case of the Lagrangian collapse. As
another corollary we disprove Y.Chekanov's conjecture about singularities of
the Lagrangian projection of certain Lagrangian tori in $R^4$.

Differential Geometry;Differential Geometry;Symplectic Geometry