Is Newton second law covariant or invariant?

Question

Is Newton second law covariant or invariant between two inertial frames, moving with uniform traslational motion with respect to each other?

If it is invariant then, indipendently from the frame, $\vec{a}=\vec{a'}$ and $\vec{F}=\vec{F'}$ (of course $m=m'$) and this means that $\vec{F}=m\vec{a}$ has exactly the same form in both the frames.

I'm totally ok with $\vec{a}=\vec{a'}$, but how to be sure that $\vec{F}=\vec{F'}$ (without using the fact that $\vec{F'}=m\vec{a'}$)?

Moreover is this true if the two frames are are oriented differently (but the orientation is constant), as showed in the picture?

Answer 1

Newton's second law is covariant, as it does not change its form if we switch to another frame of reference. As already explained by @AccidentailFourierTransform in his comment, Newton's second law is a vector law. This means the quantities in the law are vectors, which have different values in different frames of reference. Only scalars, by definition, do not change their value if we move to another frame of reference and therefore called invariant.

Answer 2

To add more details to jimjo's answer, I would like to explain the "at most" in my comment

Vector, at most, can be covariant.

Three-vectors are only covariant under rotations, but if you include boosts then three-vectors transform in a non-covariant way. Therefore, Newton's second Law is non-covariant under the full Lorentz Group. To get a covariant equation you need to add "a zeroth component" to Newton's 2nd Law: $$ \dot p^\mu=f^\mu $$ where now both $p$ and $f$ are four vectors - that is, covariant by definition. For more details, see Principle of covariance or this post of mine.