  The space of positive Lagrangians in an almost Calabi-Yau manifold is an open
set in the space of all Lagrangian submanifolds. A Hamiltonian isotopy class of
positive Lagrangians admits a natural Riemannian metric $\Upsilon$, which gives
rise to a notion of geodesics. We study geodesics of positive $O_n(\mathbb{R})$
invariant Lagrangian spheres in $n$-dimensional $A_m$ Milnor fibers. We show
the existence and uniqueness of smooth solutions to the initial value problem
and the boundary value problem. In particular, we obtain examples of smooth
geodesics of positive Lagrangians in arbitrary dimension. As an application, we
show that the Riemannian metric $\Upsilon$ induces a metric space structure on
the space of positive $O_n(\mathbb{R})$ invariant Lagrangian spheres in the
above mentioned Milnor fibers.
