# Is Newton second law covariant or invariant?

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?

• Only scalars can be invariant. Vector, at most, can be covariant. – AccidentalFourierTransform Apr 3 '16 at 11:10

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.