The Hamilton-Jacobi problem is revisited bearing in mind the consequences
arising from a possible bi-Hamiltonian structure. The problem is formulated on
the tangent bundle for Lagrangian systems in order to avoid the bias of the
existence of a natural symplectic structure on the cotangent bundle. First it
is developed for systems described by regular Lagrangians and then extended to
systems described by singular Lagrangians with no secondary constraints. We
also consider the example of the free relativistic particle, the rigid body and
the electron-monopole system.Comment: 40 page
