Galilean covariance of Newton's second law (${\mathbf{a}}^{\prime }=R\mathbf{a},{\mathbf{F}}^{\prime }=R\mathbf{F}$)

For a Galilean transformation $t^{\prime }=t+b$, ${\mathbf{x}}^{\prime }=R\mathbf{x}-\mathbf{U}t+\mathbf{a}$, with constant rotation $R$ and boost $\mathbf{U}$, velocity transforms as ${\mathbf{v}}^{\prime }=R\mathbf{v}-\mathbf{U}$ and acceleration as ${\mathbf{a}}^{\prime }=R\mathbf{a}$. With invariant inertial mass, Newton's second law retains its form only if the actual interaction laws and their physical background data transform to give ${\mathbf{F}}^{\prime }=R\mathbf{F}$. Acceleration invariance alone does not prove covariance of an arbitrary force law. For example, resistance depends on velocity relative to a medium, whose velocity must transform too.