  In this paper, we present a generalization of a Hamilton--Jacobi theory to
higher order implicit differential equations. We propose two different
backgrounds to deal with higher order implicit Lagrangian theories: the
Ostrogradsky approach and the Schmidt transform, which convert a higher order
Lagrangian into a first order one. The Ostrogradsky approach involves the
addition of new independent variables to account for higher order derivatives,
whilst the Schmidt transform adds gauge invariant terms to the Lagrangian
function. In these two settings, the implicit character of the resulting
equations will be treated in two different ways in order to provide a
Hamilton--Jacobi equation. On one hand, the implicit differential equation will
be a Lagrangian submanifold of a higher order tangent bundle and it is
generated by a Morse family. On the other hand, we will rely on the existence
of an auxiliary section of a certain bundle that allows the construction of
local vector fields, even if the differential equations are implicit. We will
illustrate some examples of our proposed schemes, and discuss the applicability
of the proposal.
