We present an unified approach to study online scheduling problems in the resource augmentation/speed scaling models. Potential function method is extensively used for analyzing algorithms in these models; however, they yields little insight on how to construct potential functions and how to design algorithms for related problems. In the paper, we generalize and strengthen the dual-fitting technique proposed by Anand et al. [1]. The approach consists of considering a possibly non-convex relaxation and its Lagrangian dual; then constructing dual variables such that the Lagrangian dual has objective value within a desired factor of the primal optimum. The competitive ratio follows by the standard Lagrangian weak duality. This approach is simple yet powerful and it is seemingly a right tool to study problems with resource augmentation or speed scaling. We illustrate the approach through the following results.