How is the local-scale gravitational instability influenced by the surrounding large-scale structure formation?

We develop the formalism to investigate the relation between the evolution of the large-scale (quasi) linear structure and that of the small-scale non linear structure in Newtonian cosmology within the Lagrangian framework. In doing so, we first derive the standard Friedmann expansion law using the a veraging procedure over the present horizon scale. Then the large-scale (qu asi) linear flow is defined by averaging the full trajectory field over a l arge-scale domain, but much smaller then the horizon scale. The rest of the full trajectory field is supposed to describe small-scale nonlinear dynami cs. We obtain the evolution equations for the large-scale and small-scale p art of the trajectory field. These are coupled each other in most general s ituations. It is shown that if the shear deformation of fluid elements is i gnored in the averaged large-scale dynamics, the small-scale dynamics is de scribed by Newtonian dynamics in an effective Friedmann-Robertson-Walker (F RW) background with a local scale factor. The local scale factor is defined by the sum of the global scale factor and the expansion deformation of the averaged large-scale displacement field. This means that the evolution of small-scale fluctuations is influenced by the surrounding large-scale struc ture through the modification of FRW scale factor. The effect might play an important role in the structure formation scenario.