When Lenny sees a particle moving on the trajectory x(t), George sees the same particle moving on the trajectory X = x(t) − f(t). If George does not want to keep asking Lenny what the trajectory is, then he wants his own laws of motion to describe the object from his coordinates. The easiest way to transform the equations of motion from one coordinate system to another is to use the principle of least action, or the Euler-Lagrange equations.
