The geometric framework for the Hamilton-Jacobi theory is used to study this theory in the ambient of higher-order mechanical systems, both in the Lagrangian and Hamiltonian formalisms. Thus, we state the corresponding Hamilton-Jacobi equations in these formalisms and apply our results to analyze some particular physical examples.
Colombo, L. [et al.]. "Geometric Hamilton-Jacobi theory for higher-order autonomous systems". 2013.