I will discuss a degenerate form of the special Lagrangian equation that arises as the geodesic equation for the space of positive Lagrangians. Considering graph Lagrangians in Euclidean space, the equation reduces to a second order fully non-linear PDE for a single real function. I will explain how to prove existence and uniqueness for Lipschitz solutions to the Dirichlet problem on a convex domain times the unit interval. The proof uses the subequation theory of Harvey-Lawson. Existence of solutions on general manifolds with sufficient regularity would imply a version of the strong Arnold conjecture in Hamiltonian dynamics as well as uniqueness for special Lagrangians. This talk is based on joint work with Yanir Rubinstein.