Intrinsic ADM formalism for generally covariant higher-order field theories

In the framework of Higher-Order Calculus of Variations in fibered manifolds, we discuss a global and intrinsic approach to ADM formalism for (higher-order) geometric field theories, i.e., Lagrangian field theories depending covariantly on geometric objects together with their derivatives of any arbitrary order. Our approach is based on a broad use of global Poincar\'{e}--Cartan forms (whose existence in the higher-order case has been recently proven) and on the energy density which can be derived from such forms. This allows to obtain directly boundary contributions to the Hamiltonian, thus providing a general construction for so-called ``modified Hamiltonians''. To provide a simple ``pedagogical'' example, it is shown how this intrinsic approach can be used to recover the well-known standard ADM formulation of Maxwell electrodynamics on a curved background, together with the natural boundary terms for the modified Hamiltonian.