Particles and strings in degenerate background fields

We consider a model that describes a non-relativistic particle immersed in a degenerate gravitational background, where we apply the Hamiltonian formalism and we obtain the constraints. According to the form of background components, we may have a system with first or second class constraint. We also study string theory in a similar background, where the Hamiltonian analysis is non-trivial because of the new structure of constraints generated in this case. (author)