On the classical confinement of test particles to a thin 3-brane in the   absence of non-gravitational forces
  The classical confinement condition of test particles to a brane universe in the absence of non-gravitational forces is transformed using the Hamilton-Jacobi formalism. The transformed condition provides a direct criterion for selecting in a cosmological scenario 5D bulk manifolds wherein it is possible to obtain confinement of trajectories to 4D hypersurfaces purely due to classical gravitational effects. 
M. La Camera