Clarification of coordinate fictitious forces

[etotheipi]

I was reading through this Wikipedia article and stumbled across a section related to outlining the differences between "state-of-motion" fictitious forces and "coordinate" fictitious forces. I have no idea what the second category is supposed to be, and wondered whether someone could explain!

E.g. in a rotating frame, you get a (coordinate system independent) fictitious force of $-m\frac{d\vec{\omega}}{dt} \times \vec{r} - 2m\vec{\omega} \times \frac{d\vec{r}}{dt} - m\vec{\omega} \times (\vec{\omega} \times \vec{r})$.

Now, on a separate note, in a polar coordinate system the acceleration is $(\ddot{r} - r\dot{\theta}^2)\hat{r} + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\hat{\theta}$

It was my understanding that in a non-inertial frame you can simply treat the fictitious forces as real forces and do everything else as you would in an inertial frame. However, for some reason, the article is suggesting that terms in the decomposition of the acceleration in different coordinate systems, like $-r\dot{\theta}^2\hat{r}$ in a polar coordinate system, are also fictitious "coordinate forces".

This makes zero sense to me; surely they're just terms pertaining to the acceleration relative to the rotating frame? Why would you call them forces - they don't even have the right dimensions?

Thanks!

 

[PeroK]

I haven't read all the Wikipedia page but imagine the following scenario. You are moving inertially (this is a physical statement that refers to whether Newton's first law is obeyed in your reference frame). But, you choose a rotating coordinate system. This is simply the way you assign coordinates to points in space and has no effect on your state of motion. Now, you have "coordinate" centrifugal and Coriolis terms appear in your equations of motion. E.g. for particles that are at rest relative to you.

This is different from the case where you are moving non-inertially - on the surface of the Earth, say, but using a Cartesian coordinate system (within yout frame of reference). In this case, the centrifugal and Coriolis terms appear because of your state of motion.

 

[etotheipi]

Ah okay, so the latter has to do with the case where the frame of reference in which the coordinate system is being set up is non-inertial, whilst the former applies in the case where the frame of reference is inertial but you choose the coordinate system in a strange way.

I'm still not sure I entirely understand the first case; shouldn't the coordinate system be at rest wrt the frame of reference? I.e. in an inertial frame we don't choose non-inertial coordinate systems, since that doesn't make any sense? That is to say that if you choose a rotating coordinate system then you are also implying a rotating frame of reference, which is a non-inertial thing.

I can only understand the case where you choose a frame of reference (inertial, non-inertial/rotating, doesn't matter), you fix a coordinate system at rest to that frame and formulate some laws of motion. This is pretty much what your example 2 is saying.

 

[A.T.]

This is different from the case where you are moving non-inertially - on the surface of the Earth, say, but using a Cartesian coordinate system (within yout frame of reference). In this case, the centrifugal and Coriolis terms appear because of your state of motion.

I don't see how this is different. They again appear because you have chosen non-inertial coordinates. The fact that you are also non-inertial and at rest w.r.t. to these coordinates is irrelevant.

 

[PeroK]

Ah okay, so the latter has to do with the case where the frame of reference in which the coordinate system is being set up is non-inertial, whilst the former applies in the case where the frame of reference is inertial but you choose the coordinate system in a strange way.

I'm still not sure I entirely understand the first case; shouldn't the coordinate system be at rest wrt the frame of reference? I.e. in an inertial frame we don't choose non-inertial coordinate systems, since that doesn't make any sense? That is to say that if you choose a rotating coordinate system then you are also implying a rotating frame of reference, which is a non-inertial thing.

I can only understand the case where you choose a frame of reference (inertial, non-inertial/rotating, doesn't matter), you fix a coordinate system at rest to that frame and formulate some laws of motion. This is pretty much what your example 2 is saying.

Whether you are moving inertially or not, you are free to use any coordinate system you want. It's natural to use a Cartesian system with a fixed origin (at rest relative to you). But, you can use any coordinates you like. Often you associate your choice of coordinates with another frame of reference: the frame of reference of the moving train; the accelerating reference frame of the ball. But, you don't have to see this as "someone else's frame". You can see this purely as your choice of coordinates.

 

[A.T.]

I was reading through this Wikipedia article and stumbled across a section related to outlining the differences between "state-of-motion" fictitious forces and "coordinate" fictitious forces. I have no idea what the second category is supposed to be, and wondered whether someone could explain!

After a short look it seems they mean the following:

Consider an inertial body moving at constant velocity. In inertial Cartesian coordinates the second derivatives of all position coordinates are zero. But in inertial polar coordinates they are not. This is neither due to the proper acceleration of the body nor due to the proper acceleration of the coordinates (both are inertial), but due to the geometry of the coordinates.

This makes zero sense to me; surely they're just terms pertaining to the acceleration relative to the rotating frame? Why would you call them forces - they don't even have the right dimensions?

All inertial forces are just coordinate accelerations, which you then multiply by mass to call them forces.

 

[PeroK]

I don't see how this is different. They again appear because you have chosen non-inertial coordinates. The fact that you are also non-inertial and at rest w.r.t. to these coordinates is irrelevant.

The two scenarios are physically different.

 

[Dale]

the differences between "state-of-motion" fictitious forces and "coordinate" fictitious forces

I don’t think that this distinction is made in any source that I have read previously. I wouldn’t worry about it. If you like it then use it, if not then don’t.

 

[etotheipi]

I wonder if you are both discussing distinct things. On the one hand, there is the issue of whether a fictitious force appears due to working in a certain non-inertial frame of reference vs instead due to defining an accelerating coordinate system within an inertial frame.

But, you don't have to see this as "someone else's frame". You can see this purely as your choice of coordinates.

I understand this point, though I'd need to think about it a little longer. I was under the impression a coordinate system was defined at rest w.r.t. a frame of reference, and that fictitious forces arise only in non-inertial frames. This sidesteps the whole issue of "coordinate" fictitious forces.

And on the other hand, there is the issue of the non-zero second derivatives. This appears equally strange to me; the acceleration vector is coordinate independent, though granted its components will depend on the choice of a coordinate system. That said, I thought inertial forces were introduced so that you make laws like $\vec{F} = m\vec{a}$ work right again. If you move coordinate accelerations to the other side of the equation and multiply by mass, then the $\vec{a}$ you're left with on the right hand side is no longer the acceleration but only part of it. I don't see how it's helpful to introduce these new coordinate inertial forces.

 

[A.T.]

The two scenarios are physically different.

Yes, but the reason for inertial forces (in the common sense) is the choice of coordinates, not the scenario.

 

[Dale]

@PeroK and @A.T. you may want to both read the Wikipedia article in question to see what @etotheipi is asking about.

 

[A.T.]

@PeroK and @A.T. you may want to both read the Wikipedia article in question to see what @etotheipi is asking about.

I did, and I still think they talk about what I described in post #6:

"state-of-motion fictitious forces": come from the non-inertial reference frame

"coordinate fictitious forces ": come from non-Cartesian coordinates (and can appear in inertial frames too.)

 

[etotheipi]

I did find this, which I hadn't seen (analysis is from an inertial frame here):

From a mathematical standpoint, however, it sometimes is handy to put only the second-order derivatives on the right side of this equation; that is we write the above equation by rearrangement of terms as:

where a "coordinate" version of the "acceleration" is introduced:

consisting of only second-order time derivatives of the coordinates r and θ. The terms moved to the force-side of the equation are now treated as extra "fictitious forces" and, confusingly, the resulting forces also are called the "centrifugal" and "Coriolis" force.

These newly defined "forces" are non-zero in an inertial frame, and so certainly are not the same as the previously identified fictitious forces that are zero in an inertial frame and non-zero only in a non-inertial frame.[38] In this article, these newly defined forces are called the "coordinate" centrifugal force and the "coordinate" Coriolis force to separate them from the "state-of-motion" forces.

So it would appear this new construction works so long as we don't confuse the actual acceleration $\mathbf{a}$ in the frame of reference with this bizarre modified $\tilde{\mathbf{a}}$ which contains only second derivative terms.

I'm not sure I get the point, but it's valid, I guess.

 

[etotheipi]

Whether you are moving inertially or not, you are free to use any coordinate system you want. It's natural to use a Cartesian system with a fixed origin (at rest relative to you). But, you can use any coordinates you like. Often you associate your choice of coordinates with another frame of reference: the frame of reference of the moving train; the accelerating reference frame of the ball. But, you don't have to see this as "someone else's frame". You can see this purely as your choice of coordinates.

I just had one last slightly more general question about this part; I thought about it for a little while but in the end couldn't come to a good enough understanding. If the chosen coordinate system wasn't at rest with respect to your frame of reference, wouldn't we need a second coordinate system that was in order to quantify the motion of that moving coordinate system in the first place?

I just thought that for a coordinate system to be useful, it needs to be at rest w.r.t. the frame of reference it's being employed in, which implies that the set of possible coordinate systems are related by constant translations and rotations of the basis vectors from one initial system fixed w.r.t. the frame of reference.

That is to say that it only makes sense to me to associate a coordinate system fixed to the train with the rest frame of the train. And if the train is accelerating, you need the inertial forces; but no matter what, you won't get any inertial forces in the frame of the inertial observer no matter their choice of coordinate system.

 

[PeroK]

I just had one last slightly more general question about this part; I thought about it for a little while but in the end couldn't come to a good enough understanding. If the chosen coordinate system wasn't at rest with respect to your frame of reference, wouldn't we need a second coordinate system that was in order to quantify the motion of that moving coordinate system in the first place?

I just thought that for a coordinate system to be useful, it needs to be at rest w.r.t. the frame of reference it's being employed in, which implies that the set of possible coordinate systems are related by constant translations and rotations of the basis vectors from one initial system fixed w.r.t. the frame of reference.

That is to say that it only makes sense to me to associate a coordinate system fixed to the train with the rest frame of the train. And if the train is accelerating, you need the inertial forces; but no matter what, you won't get any inertial forces in the frame of the inertial observer no matter their choice of coordinate system.

I was thinking perhaps that you can only physically make measurements from your frame of reference, but you can analyse the system using any coordinates. But, perhaps there isn't much difference between this and using a difference reference frame after all.

 

[Dale]

I just thought that for a coordinate system to be useful, it needs to be at rest w.r.t. the frame of reference it's being employed in,

All coordinate systems are equally useful, regardless of the motion of any objects within the coordinate system.

Some coordinate systems are more convenient than others. If a system has a certain symmetry then using a coordinate system that shares that symmetry will often simplify the equations and make things more convenient.

 

[etotheipi]

I was thinking perhaps that you can only physically make measurements from your frame of reference, but you can analyse the system using any coordinates. But, perhaps there isn't much difference between this and using a difference reference frame after all.

It's sort of an interesting question, a little confusing to think about! I think you're right, a coordinate system moving w.r.t. an observer in a given reference frame would be equivalent to just having one at rest in another reference frame.

I remember the thread a while ago about trying to iron out some of this terminology and I recall the Wikipedia page for reference frames having some good definitions on it. I just went to see and one mentions that for a chosen coordinate system in a given frame

The corresponding set of axes, sharing the rigid body motion of the frame $\mathcal{R}$ can be considered to give a physical realization of $\mathcal{R}$

So I suspect it's just more of a definition than anything else.

 

[etotheipi]

All coordinate systems are equally useful, regardless of the motion of any objects within the coordinate system.

Some coordinate systems are more convenient than others. If a system has a certain symmetry then using a coordinate system that shares that symmetry will often simplify the equations and make things more convenient.

Right, but so long as the origin of that system is at rest with respect to the frame of reference? Polar coordinates exploit symmetry but still need to have the pole at rest in the frame.

 

[Dale]

Right, but so long as the origin of that system is at rest with respect to the frame of reference? Polar coordinates exploit symmetry but still need to have the pole at rest in the frame.

No. That is not necessary. You can do all of the physics using any coordinate system you like.

 

[etotheipi]

No. That is not necessary. You can do all of the physics using any coordinate system you like.

But then I wonder, what is the need to even define a frame of reference? In special relativity I've only ever seen "frame S' moving relative to S", and not "two coordinate systems A and B defined in frame S".

And furthermore, we'd need to allow for the possibility of fictitious forces in our equations due to the possibility of an accelerating coordinate system, even though the frame of reference is inertial.

Sure - it's doable - but surely at that point you might as well just call them different frames of reference and let all coordinate systems be fixed w.r.t. different frames?

 

[hutchphd]

May I also point out that calling anything "fictitious" gives you a certain latitude as to its definition! As long as the dimensions are correct, that would imply carte blanche to me...

 

[etotheipi]

Or to put it another way, the definition of an inertial reference frame is one in which Newton's law of inertia holds with only real forces. If we setup an accelerating coordinate system within an IRF then Newton's law of inertia no longer holds, so it would not be an IRF - this is a contradiction.

Wikipedia tells me that

In physics, a frame of reference (or reference frame) consists of an abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurements within that frame.

and later on that

However, a mere shift of origin, or a purely spatial rotation of space coordinates results in a new coordinate system. So frames correspond at best to classes of coordinate systems

I'm not too sure what I am overlooking here; surely it is reasonable to deduce that systems of coordinates are at rest w.r.t. a frame of reference? Of course, the type of coordinate system doesn't matter.
On another note, happy Friday! And thanks for your help!

 

[A.T.]

If we setup an accelerating coordinate system within an IRF

That doesn't make any sense. A inertial reference frame is a set of coordinate systems that are all related by time-independent transformations (at rest relative to each other), so they are all inertial.

 

[etotheipi]

That doesn't make any sense. A inertial reference frame is a set of coordinate systems that are all related by time-independent transformations (at rest relative to each other), so they are all inertial.

Yes, that is my view. That possible coordinate systems are related by constant translations and rotations and consequently all coordinate systems are at rest relative to the frame of reference they are set up in.

 

[A.T.]

Yes, that is my view. That possible coordinate systems are related by constant translations and rotations and consequently all coordinate systems are at rest relative to the frame of reference they are set up in.

And all these coordinate system are non-accelerating. They have no "state-of-motion fictitious forces". They can only have "coordinate fictitious forces" if non-Cartesian, but that doesn't imply proper acceleration of the coordinates.

 

[etotheipi]

I assume it is also correct to say that a non-inertial frame is still a set of coordinate systems related by time-independent transformations. But the distinction just lies in whether Newton's first law holds in those coordinate systems.

And all these coordinate system are non-accelerating. They have no "state-of-motion fictitious forces". They can only have "coordinate fictitious forces" if non-Cartesian, but that doesn't imply proper acceleration of the coordinates.

Yeah. I don't think I'll ever use this idea of "coordinate fictitious forces" since I don't think it's a particularly useful concept. I'm not too sure why it would ever be beneficial to have the double derivative terms on the RHS. In fact, it's just more confusing since it actually breaks Newton's second law instead of fixing it (since we now need to use a modified acceleration $\tilde{\mathbf{a}}$ which differs from the actual acceleration as measured by that frame). Thank you!

 

[Dale]

But then I wonder, what is the need to even define a frame of reference? In special relativity I've only ever seen "frame S' moving relative to S", and not "two coordinate systems A and B defined in frame S".

In discussions about relativity very often reference frames and coordinate charts are treated as synonyms, and for the most part there is no harm and little ambiguity. But technically they are different things.

A coordinate chart is a smooth one-to-one mapping between events in spacetime and points in R4. A frame field (aka tetrad or vielbein) is a set of four orthonormal vectors defined at each event, one timelike and three spacelike.

These are different concepts. For example, a reference frame does not include any notion of simultaneity. Therefore, you can easily make a perfectly physically sensible tetrad for observer’s on a rotating disk even though coordinate systems all have difficulty with physically reasonable synchronization.

From a coordinate chart it is always possible to obtain coordinate basis vectors, but those coordinate basis vectors may not form a proper tetrad. So coordinate charts and frame fields are distinct concepts.

 

[etotheipi]

These are different concepts. For example, a reference frame does not include any notion of simultaneity. Therefore, you can easily make a perfectly physically sensible tetrad for observer’s on a rotating disk even though coordinate systems all have difficulty with physically reasonable synchronization.

Interesting! Yes my earlier post was in the vein that if we were to allow coordinate systems to translate/rotate in a time independent manner w.r.t. a reference frame (inertial or non-inertial), then there would be little point to reference frames since we could just pick one, stick to it and define all of our coordinate systems in that frame.

Once you make the stipulation that @A.T. gave a few posts back, the differences between them become more evident. For instance, coordinate systems translating relative to each other occupy different rest frames, and we now have the relativity of simultaneity to worry about if we want to transform between them!

 

[A.T.]

Yeah. I don't think I'll ever use this idea of "coordinate fictitious forces" since I don't think it's a particularly useful concept.

An interesting case are non-inertial non-Cartesian coordinates, where the "coordinate fictitious forces" exactly cancel the "state-of-motion fictitious forces", so you get rid of all the "fictitious forces" and model non-inertial frames using geometry. This is related to how gravity is modeled in General Relativity.

 

[etotheipi]

Those animations are awesome! It's been a while since I read about the equivalence principle and I didn't get too far at all into the details so I'm quite rusty, so I might go and brush up on that.

I guess I spoke too soon about the "coordinate fictitious forces"...