Imagine we´re dealing with a rigid body and we want to apply for it lagrangian mechanics, in order to obtain its equations of motion. If we want to compute the generalized force associated to a specific generalized coordinate, I´ve observed that, when gravity is the only force, and we consider gravitational field to be uniform in the region, generalized force can be written as the total force on the body times the displacement of the center of mas with respect to the generalized coordinate.
It would be equivalent to consider the rigid body to be a point mass at which the total force on the body is acting, which is located at the center of mass of the rigid body, and whose mass equals the total mass of the rigid body.
Why is this possible? Would it be extensible to non-uniform forces?