As in Landau & Lifshitz 1st chapter
"If an inertial frame К is moving with an infinitesimal velocity e relative to another inertial frame K', then v' = v+e. Since the equations of motion must have the same form in every frame, the Lagrangian L(w2) must be converted by this transformation into a function L' which differs from £(w2), if at all, only by the total time derivative of a function of co-ordinates and time We have U = L(v'2) = L(v2 + 2v • e + e2)."
Could we use the same arguments to find lagrangian of free particle in non-inertial frame transforming frame using $$r'=ut+(at^2)/2~?$$
