Let T(S) be a Teichmaller space of a hyperbolic Riemann surface S, viewed as a set of Teichmaller equivalence classes of Beltrami differentials on S. It is shown in this paper that for any ex-tremal Beltrami differential μo at a given point T of T( S), there is a Hamilton sequence forμo formed by Strebel differentials in a natural way. Especially, such a kind of Hamilton sequence possesses some special properties. As applications, some results on point shift differentials are given.