In this work we establish that disconjugacy of a linear Hamiltonian system on time scales is a necessary condition for the positivity of the corresponding quadratic functional. We employ a certain minimal normality (controllability) assumption. Hence, the open problems stated by the author in [15], [16] are solved with the result that positivity of the quadratic functional is equivalent with disconjugacy of the Hamiltonian system on the interval under consideration. The general approach on time scales T involves, as special cases, the well known continuous case for T=R and recently developed discrete one for T=Z, so that they are unified. As applications, Sturmian type separation and comparison theorems on time scales are also provided.