  We study the rank stratification for the differential of a Lagrangian
fibration over a smooth basis. We also introduce and study the notion of
Lagrangian morphism of vector bundles. As a consequence, we prove some of the
vanishing, in the Chow groups of a Lagrangian fibered hyper-K\"ahler variety
$X$, of certain polynomials in the Chern classes of $X$ and the Lagrangian
divisor, predicted by the Beauville-Voisin conjecture. Under some natural
assumptions on the dimensions of the rank strata, we also establish
nonnegativity and positivity results for Chern classes.
