For each simple Lie algebra g (excluding, for trivial reasons, type C) we find the lowest possible degree of an invariant 2 nd order PDE over the adjoint variety in Pg, a homogeneous contact manifold. Here a PDE F (x i , u, u i , u ij ) = 0 has degree ≤ d if F is a polynomial of degree ≤ d in the minors of (u ij ), with coefficients functions of the contact coordinates x i , u, u i (e.g., Monge-Ampère equations have degree 1). For g of type A or G 2 we show that this gives all invariant 2 nd order PDEs. For g of type B and D we provide an explicit formula for the lowest-degree invariant 2 nd order PDEs. For g of type E and F 4 we prove uniqueness of the lowest-degree invariant 2 nd order PDE; we also conjecture that uniqueness holds in type D.