DrStupid said:
What has that to do with inertial frames? The laws of motion (with the wording above) are not valid in all frames of reference. In #85 I demonstrated how rotating frames violate them. They can either comply with Law I and II or with Law III but not with all of them at once.
From post #85:
DrStupid said:
Newton I: As the particle remains ar rest, there is no force acting on it.
Newton II: As the acceleration is zero, the force acting on the particle is ##F = m \cdot a = 0##
Now I switch to another frame of reference that is rotating around the origin with the angular velocity ##\omega##. In this frame the particle is moving on circular paths around the rotational axis. That means according to
Newton I: As the particle doesn't remain at rest or uniform translation, there is a force acting on it.
Newton II: As the acceleration is ##- \omega ^2 \cdot r##, the force acting on the particle is ##F = m \cdot a = - m \cdot \omega ^2 \cdot r##
But there is a catch. I think nothing in your (or Newton's) formulation of the axioms implies that switching to a rotating frame changes the particle's acceleration to ##-\omega^2 r##. You probably think that you are just using standard definitions here, but you are not. It is your definition of "acceleration" that only applies to inertial frames, not any of Newton's axioms.
I suppose we agree that acceleration is just the time derivative of velocity, both of which are represented as vectors. The velocity can be calculated as time derivative of position, which may also be represented as vector relative to the common origin. Now we just have to carefully apply those definitions: Since both the inertial frame S, ##\boldsymbol{e}_i##, and the rotating frame S', ##\boldsymbol{\tilde e}_i## always share the same origin, the position vector ##\boldsymbol{r}## at every instant of time is given by (left hand side is always S, right hand side is S' in the following)
$$ r^i\boldsymbol{e}_i = \boldsymbol{r} = \tilde r^i\boldsymbol{\tilde e}_i.$$
Its time derivative gives the velocity
$$\dot{r}^i \boldsymbol{e}_i = \boldsymbol{v} = \dot{\tilde r}^i\boldsymbol{\tilde e}_i + \boldsymbol{\omega}\times\boldsymbol{r}$$
You assumed ##\dot{r}^i \equiv 0##, so ##\boldsymbol{v}=0##.
Note that the velocity w.r.t
both systems is just ##\boldsymbol{v}=0##. It is
not given by ##\dot{\tilde r}^i\boldsymbol{\tilde e}_i## in the rotating system, because that vector simply isn't equal to the time derivative
$$\boldsymbol{v} = \frac{\mathrm{d}\boldsymbol{r}}{\mathrm{d}t},$$
which I just defined as velocity. Although velocity is
relative to the frame of reference, this relativity only produces a
difference if both origins are in relative motion, which they are not in the situation you described. (If you think of it, it really makes sense that way. Relative velocity is just a vector in absolute space. It may depend on time, but why would it depend on an arbitrary frame, that someone sets up in
space? A vector, as a geometric object, is supposed to be independent of basis.)
Now proceeding to acceleration, using ##\boldsymbol{v}=0##, ##\boldsymbol{\omega}=\text{const.}##, and -- obviously -- ##\boldsymbol{r}\cdot\boldsymbol{\omega}=0##, we get
$$ \ddot{r}^i\boldsymbol{e}_i = \boldsymbol{a} = \ddot{\tilde r}^i \boldsymbol{\tilde e}_i + \dot{\tilde r}^i \boldsymbol{\omega}\times \boldsymbol{\tilde e}_i = \ddot{\tilde r}^i \boldsymbol{\tilde e}_i - \boldsymbol{\omega}\times(\boldsymbol{\omega}\times\boldsymbol{r}) = \ddot{\tilde r}^i\boldsymbol{\tilde e}_i + \omega^2 \boldsymbol{r}$$
which again vanishes since the left hand side vanishes by assumption. Now it is clear that what you call "acceleration" is just the term
$$\ddot{\tilde r}^i\boldsymbol{\tilde e}_i = -\omega^2 \boldsymbol{r}.$$
But again, this is
not the time derivative of velocity relative to the rotating system, because it neglects the rotation of the axes, which change with time. It is no accident that textbooks introduce a special notation for this kind of operation, usually
$$\left(\frac{\mathrm{d}^2\boldsymbol{r}}{\mathrm{d}t^2}\right)_{S'}$$
because it needs to be distinguished from the covariant time derivative ##\mathrm{d}/\mathrm{d}t##.
Now another point needs to be mentioned. Just as the relative velocities are equal, iff the origins are not in relative motion, so the accelerations -- calculated in the way above -- are equal only if the origins have constant relative velocity. Now, from what I just said it seems to follow that the application of Newton's axioms will be ambiguous if both origins are accelerating relative to each other. Because in that situation both systems seem to assign different accelerations to the same particle. But this is only apparent. Let's review the basic assumption that real forces, according either to the Third Law or to our general understanding of the interactions of that particular system, represent the cause for change of the particle's state of motion. Any causal effects on this state can only depend on the particle's state and properties, not on how arbitrarily definable "origins" are moving.
This is why the acceleration in ##m\boldsymbol{a} = \boldsymbol{F}## must be covariantly defined, by
$$\nabla_{\boldsymbol{u}} \boldsymbol{u} = \boldsymbol{a},$$
which manifestly depends only on the particles motion through spacetime. This acceleration equals the second derivative of position ##\frac{\mathrm{d}^2 \boldsymbol{r}}{\mathrm{d}t^2} = \nabla_{\boldsymbol{e}_0}\boldsymbol{v} = \nabla_{\boldsymbol{e}_0}(\boldsymbol{u} - \boldsymbol{e}_0) = \boldsymbol{a} - \nabla_{\boldsymbol{e}_0} \boldsymbol{e}_0## only under the condition that ##\nabla_{\boldsymbol{e}_0}\boldsymbol{e}_0 = 0##, i.e.
if the origin itself is moving on a straight and uniform trajectory (geodesic). This should be an immediately intuitive result.
That a definition based on the (second order) change of position can only be correct in special situations is, I think, not surprising. "Change of position" is an ill-defined concept anyway in light of the Principle of Relativity. It is only defined relative to an object of reference, which can itself change its state of motion arbitrarily. What objects are suitable references for this purpose is completely explained within the theory by means of the analysis above. What is always well-defined, however, without any reference are deviations from geodesic (straight and uniform) motions in spacetime.
In elementary treatments this complication doesn't show up, because there is always one inertial system involved relative to which all accelerations can be unambiguously equated with the second derivative of position (which is therefore used as a universal
definition of acceleration). This makes it less obvious how a fully covariant formulation looks like and it seems that one inertial frame is always required. That this is not so, is only apparent when the laws are formulated in Newtonian spacetime. In any case, as shown by the Newton-Cartan-Theory, it is
possible to formulate Newton's laws in a fully covariant way, which means the claim that some of these laws
must be violated in non-inertial reference frames seems untenable to me. Furthermore, this spurious violation cannot be a valid basis for the definition of inertial frames.