Abstract: In this paper,we study the global smooth solutions of the Cauchy problem for two important nonstrictly quasilinear hyperbolic systems.i.e.,the isentropic gas dynamics system in Euler coordinates (Ⅰ) and the rotational degeneracy of hyperbolic systems of conservation laws(Ⅱ).sufficient conditions which guarantee the existence of gloats smooth solutions of the Cauchy problems (Ⅰ) and (Ⅱ) are obtained by employing the characteristic method.

Key words: Nonstrictly quasilinear hyperbolic system, a priori estimates, global smooth solution