In this work we present a unified treatment of continuous and discrete vector linear Hamiltonian systems on a general time scale T, with the matrix $B$_t not necessarily invertible. This contains as special cases the Sturm-Liouville differential and difference equations of higher order. We define generalized zeros for vector solutions $(x,u)$ of the Hamiltonian system. From this we read off the corresponding definition of generalized zero points for solutions of the Sturm-Liouville equations, so that the well known continuous case (T=R) and recently developed discrete one (T=Z) are unified. We show that disconjugacy of the equation implies positivity of the corresponding quadratic functional.

Last change: August 28, 2000. (c) Roman Hilscher