  We study the Hamiltonian formulation for a parametrized scalar field in a
regular bounded spatial region subject to Dirichlet, Neumann and Robin boundary
conditions. We generalize the work carried out by a number of authors on
parametrized field systems to the interesting case where spatial boundaries are
present. The configuration space of our models contains both smooth scalar
fields defined on the spatial manifold and spacelike embeddings from the
spatial manifold to a target spacetime endowed with a fixed Lorentzian
background metric. We pay particular attention to the geometry of the infinite
dimensional manifold of embeddings and the description of the relevant
geometric objects: the symplectic form on the primary constraint submanifold
and the Hamiltonian vector fields defined on it.
