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Inspired by problems arising in the geometrical treatment of Yang-Millstheories and Palatini's gravity, the covariant formulation of Hamiltoniandynamical systems as a Hamiltonian field theory of dimension $1+0$ on amanifold with boundary is discussed. After a precise statement of Hamilton'svariational principle in this context, the geometrical properties of the spaceof solutions of the Euler-Lagrange equations of the theory are analyzed. Asufficient condition is obtained that guarantees that the set of solutions ofthe Euler-Lagrange equations at the boundary of the manifold, fill a Lagrangiansubmanifold of the space of fields at the boundary. Finally a theory ofconstraints is introduced that mimics the constraints arising in Palatini'sgravity.

Covariant Hamiltonian first order field theories with constraints on manifolds with boundary: the case of Hamiltonian dynamics