 Solution

Showing that acceleration is unchanged by a uniform boost is necessary, but it is not enough to establish the Galilean covariance of Newton's second law. One must specify how time, positions, masses, forces and any background physical systems transform, and show that the actual force laws retain their form.

For a constant spatial rotation $R$, boost velocity $\mathbf{U}$ and translations, writeDifferentiation gives ${\mathbf{v}}^{\prime }=R\mathbf{v}-\mathbf{U}$ and ${\mathbf{a}}^{\prime }=R\mathbf{a}$. Assuming invariant inertial mass,Consequently the equation has the same form if the resultant physical force satisfies ${\mathbf{F}}^{\prime }=R\mathbf{F}$. For a pure boost, this reduces to ${\mathbf{F}}^{\prime }=\mathbf{F}$. More explicitly, if a force is represented by a function of positions, velocities and other physical data, those arguments and data must transform so that its primed value is $R$ times the old value. It is this condition on the complete dynamical model, not the typography of $\mathbf{F}=m\mathbf{a}$, that establishes covariance.

An instantaneous inverse-square interaction depending on the separation of two particles passes this test: their common boost cancels from the separation, and a rotation rotates the force vector. A resistance law $\mathbf{F}=-\zeta (\mathbf{v}-\mathbf{u})$ also passes when the medium's velocity $\mathbf{u}$ transforms with the body. In contrast, keeping $\mathbf{F}=-\zeta \mathbf{v}$ in every boosted frame while silently resetting the same medium to rest changes the physical problem. That model has not been shown covariant by noting ${\mathbf{a}}^{\prime }=\mathbf{a}$.

Likewise, the full Maxwell equations are not made Galilean covariant merely because some equation of particle motion is written as force equals mass times acceleration. The covariance of a framework has to be checked together with its interaction laws. Newton's own Corollary V expresses the corresponding equivalence of uniformly translating, nonrotating systems for the mechanics under consideration.^[ref]