The dynamics of polar oscillations of compact stars are conventionally described by a set of coupled differential equations and the roles that space-time and matter play in such oscillations are consequently difficult to track. In this paper, we develop a scheme for polar oscillations in terms of one space-time and one matter variables. With our judicious choice of variables, the coupling between space-time and matter is weak, which enables the separation of these two parties. Two independent second-order ordinary differential equations are obtained, corresponding to the Cowling and inverse-Cowling approximations, and leading to accurate determination of p- and w-mode quasi-normal modes, respectively, in a unified framework.