Are fictitious forces necessary to solve certain problems?

[A.T.]

[*]Model the weather to the extent that you can predict the landfall of a hurricane or the likelihood of a tornado outbreak with some reasonable semblance of accuracy.

A similar but simpler example is mesh based computational fluid dynamics for a rotating part (propeller. turbine, etc). In the non rotating frame the fluid volume has to be remeshed in each step, because the part moves. This is computationally very expensive. In the co-rotating frame the fluid spins, while the part & volume mesh are static. The CFD equations are simply modified by the inertial force terms.

I don't even want to start to imagine modelling weather in an inertial frame, that doesn't rotate with the Earth.

 

[harrylin]

A similar but simpler example is mesh based computational fluid dynamics for a rotating part (propeller. turbine, etc). In the non rotating frame the fluid volume has to be remeshed in each step, because the part moves. This is computationally very expensive. In the co-rotating frame the fluid spins, while the part & volume mesh are static. The CFD equations are simply modified by the inertial force terms.

I don't even want to start to imagine modelling weather in an inertial frame, that doesn't rotate with the Earth.

That sounds plausible. Anyway, the purpose here is to put such generic claims to the test. And different from your post #17 where you ask the claimant to provide both calculations, with and without fictitious force, I only ask for a single calculation example with fictitious force.

 

[A.T.]

The claim that I cited (of which you only retained a snippet) was that without using fictitious forces you even get an erroneous result if you do your dynamics in the inertial frame and then transform the result to the rotating frame

No. That is not what it says. Here is the quote again. I bolded the parts, that you keep ignoring:

"You have to use the fictitious forces in the rotating frame, otherwise you get the wrong trajectory. It doesn't matter if you do your dynamics in the inertial frame and then transform the result to the rotating frame or if you do your dynamics in the rotating frame directly. Either way the trajectory in the rotating frame has coordinate accelerations that are not attributable to the real forces and require fictitious forces to explain."

It specifically says that you can use both frames. But the coordinate accelerations measured in rotating frame can only be attributed to inertial forces.

 

[harrylin]

No. That is not what it says. Here is the quote again. I bolded the parts, that you keep ignoring:

It specifically says that you can use both frames. But the coordinate accelerations measured in rotating frame can only be attributed to inertial forces.

I already highlighted the parts that you keep ignoring; the bold parts do not change them. The coordinate accelerations can be attributed to the rotation of the frame, just as Newton did and GPS does.

But I let myself be side-tracked: all this talking in the air is contrary to the purpose of this thread. I asked for a calculation example with fictitious forces that we can put to the test; if you don't have one, please abstain from responding.

 

[A.T.]

The coordinate accelerations can be attributed to the rotation of the frame,

Only in trivial cases with no other forces acting. Inertial forces account for the non-inertial frame in a more general way that allows to combine them with interaction forces. The coordinate acceleration is the result of the net force, which in general is the sum of both: interaction and inertial forces.

 

[D H]

Mentor note:
Please keep future side discussions on Studiot's problem in [thread=531470]Studiot's thread[/thread].[/color]

 

[harrylin]

The book that Studiot discusses in the other thread (Dynamics 4.2) compares a calculation with and without fictitious force of a problem that is more complex than Hootenanny's rabbit; both approaches lead to the same equation and the derivations are essentially the same.
It was nice to see a first example of such a comparison, but what I'm after is to see example derivations or calculations that considerably differ in complexity.

 

[D H]

I don't even want to start to imagine modelling weather in an inertial frame, that doesn't rotate with the Earth.

Exactly. Weather models are essentially big CFD models. I couldn't imagine doing CFD with a movable mesh. What a mess!

The same applies for several of the other problems I supplied. The forward kinematics problems (the articulated robot with three or more joints) are hairy enough when the links and joints are described in non-inertial coordinates (more on this later). The same goes for a pseudo orbit about the (unstable) L1 point. Simulate such a pseudo orbit in inertial coordinates and the simulation will lose all precision in short order.

The two problems regarding the rotation of a rigid and non-rigid body introduce a new complexity. rotational problems in high school and freshman physics are very specially constructed so as to avoid a very nasty complication of rotational dynamics. In those simple problems, a rotation of 90 degrees followed by another rotation of 90 degrees is equal to a rotation of 180 degrees. This is not the case in general for rotations in 3 dimensional cartesian space. Rotations in 3D space and higher are not additive. 3D rotations don't live in ℝ³, the three dimensional cartesian space. They instead live in SO(3) (wiki article: http://en.wikipedia.org/wiki/Rotation_group), a non-commutative Lie group.

The Lie group SO(3) is hard enough to deal with when the inertia tensor is constant. Adding a time-varying inertia tensor makes for a bear of a problem. The inertia tensor for a rigid body is constant in a frame that rotates with the body. The inertia tensor for anything but a body with a spherical mass distribution is a time-varying beast when expressed in inertial coordinates. There is one word that fully describes attempting to deal with these kinds of rotational problems in inertial coordinates: Yech!

Back to the robotic arm: Here we have links and joints that are moving and rotating. Each one lives in the rather nasty space ℝ³×SO(3). A kinematic chain of such joints and links that describes the robot as a whole is the really nasty space (ℝ³×SO(3))^N. To paraphrase A.T., "I don't even want to start to imagine modeling such a system in an inertial frame."

 

[harrylin]

Thanks for the clarifications! Although it does sound plausible, so far nobody has provided a calculation example that we can put to test. As a reminder, the request of my original post:

I invite those who agree with such claims to give specific examples here. Then we can put such claims to the test by trying to do similar calculations without fictitious forces.

Not even a link to a website with such a calculation, or a reference to a paper with such a calculation?

 

[A.T.]

Thanks for the clarifications! Although it does sound plausible, so far nobody has provided a calculation example that we can put to test.

Test what?

 

[harrylin]

Test what?

Example calculations that make use of fictitious forces, and which are claimed to be much harder without such. That's what I requested in my first post.

So far there has been one calculation example with and without such forces in which it makes no difference at all, and allusions to more complex examples with fictitious forces.

Again: as nobody seems to have one ready, has someone at least a link to a website with such a calculation, or a reference to a paper with such a calculation?

 

[cmb]

A rabbit of mass 1kg is sat still at the very edge of a circular roundabout, which has a radius of 1m. The roundabout is rotating at 1 rad/s. Compute the net force acting on the rabbit from the rabbit's point of view.​

the point still remains: if you fail to account for the fictitious centrifugal force acting on the rabbit, then then net force will not vanish.

I don't see what is tricky about it at all:

At time t, the rabbit has rotated around to the northern most point of the roundabout. He looks at his compass and says "I sense an acceleration, and I determine that it is an acceleration to the North. I don't know what it is, I'll call it 'gravity B'. I also determine that I am not moving relative to the thing I am sitting on, which feels 'grippy' and seems to be holding me put, therefore I conclude there is a force between me and what I am sitting on preventing me accelerating north. I will call this a 'frictional force' holding me from accelerating, and this force is pointing south. If I hold a 100g carrot then I detect a 'gravity B' force on it of 0.1N, therefore I conclude the acceleration due to gravity B is 0.1m/s/s, so the frictional force opposing the gravity B acceleration on my 1kg mass is 1N."​

This clever rabbit has concluded there is a 1N frictional force acting on him, vectored south, as time t. (which opposes an acceleration, 'gravity B')

We examine the roundabout from our non-rotating frame:

Objects held at the edge of this roundabout are being accelerated towards its axis by w.w.r=1m/s/s. Therefore, for objects to satisfy the equation of motion whilst in circular motion about the axis, there must be a force causing this acceleration. At time t we observe a rabbit clinging to the roundabout, and has reached the northern most point of the roundabout. He is being accelerated towards the centre, because all such objects at the edge are being accelerated to the centre. The only force on him is the friction between him and the roundabout. To satisfy the equation of motion, the force must equal this acceleration, so F=1.[1^2].1=1N which, at time t, is acting south.​

The clever observer has concluded there is a 1N frictional force acting on the rabbit, vectored south, as time t. (which causes an acceleration towards the axis of rotation)



So, the net forces observed are the same. It is the perception of the acceleration that is different, just as you would expect between a non- and an accelerating frame.

(Edit; note to be precise; net forces cannot 'vanish' because there is an acceleration, therefore there is an 'equation of motion', not 'equation of forces')

 

[cmb]

I invite those who agree with such claims to give specific examples here. Then we can put such claims to the test by trying to do similar calculations without fictitious forces.

Nobody has said that in theory one could not solve such systems without the use of inertial forces and torques.

s'funny, I could have sworn that Hootenany said it..

Nevertheless I assume that calculations with fictitious forces can always be reconverted in similar calculations without them

I have to disagree with you there.

See post #3.


I don't think there is any challenge from the OP that creating fictitious forces never helps, but he has come under the impression that some folks say it is not possible to do such calculations.

I used to get my knucles wrapped by my Mathematics master at school for creating fictitious forces to make problems easy. There are many places that it can be done when dealing with complex energy interactions too, not just 'Coriolis', 'Centrifugal' and 'gravity', but I was always told to write out a full equation of motion with forces acting on masses.

 

[A.T.]

So, the net forces observed are the same. It is the perception of the acceleration that is different,

If net forces where the same, the acceleration would be the same. "Perception of the acceleration" is vague gibberish. Do you mean "coordinate acceleration"?

just as you would expect between a non- and an accelerating frame.

But I don't want to expect anything. The whole point of inertial forces is being able to assume F_net = m * a, just as if it was an inertial frame.

 

[cmb]

If net forces where the same, the acceleration would be the same. "Perception of the acceleration" is vague gibberish. Do you mean "coordinate acceleration"?

I don't understand your vague gibberish. The accelerations ARE the same - just because the perceived acceleration is different in two frames doesn't mean the rabbit flies off in two directions!?

It is the same acceleration, but how you describe that acceleration is frame-dependent.

just as you would expect between a non- and an accelerating frame.

But I don't want to expect anything. The whole point of inertial forces is being able to assume F_net = m * a, just as if it was an inertial frame.

Too bad! If you put yourself in a rotating frame, that's just the way it is.

 

[A.T.]

...perceived acceleration...

What is "perceived acceleration"?

It is the same acceleration, but how you describe that acceleration is frame-dependent.

"Proper acceleration" is frame invariant. "Coordinate acceleration" is frame-dependent. The concept of inertial forces allows to use Newtons 2nd Law, in respect to coordinate acceleration.

Too bad!

It's not bad at all.

 

[D H]

\It is the same acceleration, but how you describe that acceleration is frame-dependent.

No, it isn't. In Newtonian mechanics, the acceleration of some object will be the same in all inertial frames. Allow non-inertial frames and acceleration is no longer frame invariant. A simple example: The acceleration of a helicopter hovering over some point on the surface of the Earth is identically zero from the perspective of a frame rotating with the Earth but is non-zero from the perspective of an inertial frame.

 

[cmb]

What is "perceived acceleration"?

Close your eyes. How are you perceiving your weight? Is it an acceleration, or is it an invisible spooky force? How could you tell objectively if you were in a closed environment?

That's what I mean. The rabbit perceives this acceleration as 'gravity B'. This kind of thing - that objects experience a gravity-like force wrt some frame of their own - is common to most instances of fictitious forces. Sorry if my vocabulary crosses yours, it is not intentional to confobulate you with terms you use differently.

"Proper acceleration" is frame invariant. "Coordinate acceleration" is frame-dependent.

Then my term 'perceived acceleration' is simply your 'co-ordinate acceleration'.

The concept of inertial forces allows to use Newtons 2nd Law, in respect to coordinate acceleration.

Problem is, it is the wrong use of Newton's 2nd law. You cannot claim to use the 2nd law (acceleration is proportional to force) if the whole framework of co-ordinates is, itself, rotating. I can hear my maths master saying it now; 'WRONG! You HAVE TO write the equation of motion.'

Newton's 2nd law specifically refers to the notion of a linear function "Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur". (I don't think we need to speak latin to figure out the meaning of this!)

 

[cmb]

No, it isn't. In Newtonian mechanics, the acceleration of some object will be the same in all inertial frames. Allow non-inertial frames and acceleration is no longer frame invariant. A simple example: The acceleration of a helicopter hovering over some point on the surface of the Earth is identically zero from the perspective of a frame rotating with the Earth but is non-zero from the perspective of an inertial frame.

It is the same acceleration in a frame when referenced in the right line (of an imposed force) -- "lineam rectam (qua vis illa imprimitur)".

My point that it is not right to both cite Newton and also claim 'non-inertial accelerations' in rotational frames. There must be a force on the helicopter to maintain its relative position with respect to another accelerating body. I don't see how we are disagreeing, this is just terminologies.

 

[A.T.]

You cannot claim to use the 2nd law (acceleration is proportional to force) if the whole framework of co-ordinates is, itself, rotating.

Sure I can, if I assume inertial forces. The coordinate accelerarion in the rotating frame is proportional to the sum of all forces (interaction and inertial forces)

Newton's 2nd law specifically refers to the notion of a linear function

That might have been Newton's initial idea. But that concept was extended to non-inertial frames by others (d'Alembert, Corriolis etc.) by introducing inertial forces. The concept is still called Newton's 2nd Law even if it is more general than Newtons initial idea.

 

[cmb]

I haven't a single contention with you in regards constructing the procedures for handling non-inertial frames. But if we're building on what has gone on since Newton, better to say 'according to the d'Alembert Principle' rather than 'by Newton's 2nd law', no?

The OP's question, "are fictitious forces necessary for some problems", is quite clearly NO. But like any mathematical transformation, they may well make a problem a whole lot easier.

...And before we go back to the 'how could weather predictions do without using Coriolis forces', I'd cheekily point out that they hardly seem to qualify as 'accurate' with it!

 

[D H]

My point that it is not right to both cite Newton and also claim 'non-inertial accelerations' in rotational frames.

Your point assumes that physics hasn't changed since Newton's time. It has. Your point is invalid.

What we call Newtonian physics today is radically different from that set forth in Newton's Principia. Newton's reasoning was highly geometric; the Newtonian mechanics of today is highly algebraic. Newton's calculus is quite different from the calculus as taught and used today. Newton didn't have vectors; we do. Newton didn't have a solid concept of what constitutes a frame of reference and thus couldn't generalize that concept to non-inertial frames; we do.

There must be a force on the helicopter to maintain its relative position with respect to another accelerating body. I don't see how we are disagreeing, this is just terminologies.

This is much more than a disagreement over terminologies. You are unwilling to accept the validity of non-inertial frames. Conceptually, all frames of reference are equally valid.

However, just because all frames are equally valid conceptually does not mean that all frames are equally valid in practice. Given some problem to be solved, some frames will be much easier to work in than others, and some frames will yield much better accuracy than will others. Which frame is easiest to work in and which will yield the greatest accuracy depends very much on the problem at hand.

 

[cmb]

Your point assumes that physics hasn't changed since Newton's time. It has. Your point is invalid.

What we call Newtonian physics today is radically different from that set forth in Newton's Principia.

This seems to have become an argument picking me up on the way I am saying stuff, not on the conclusions to be reached wrt OP's question.

I mean, imagine it if you were actually right; DH; "Your point assumes that physics hasn't changed since Newton's time. It has."! Now you'll argue that 'physics' is the 'human construction', not really the actual way things work in the universe. And so we debate words, and words, and words on words, but not the actual physics, so I will exit, promtly, at this point...

 

[D H]

I mean, imagine it if you were actually right; DH; "Your point assumes that physics hasn't changed since Newton's time. It has."! Now you'll argue that 'physics' is the 'human construction', not really the actual way things work in the universe.

That is exactly right. Physics, and all the sciences, are a human construction. The goal of science is to perfectly describe "the actual way things work in the universe." Our current knowledge is not perfect. There is, and will probably always remain, room for improvement. The goal of perfectly describing reality is to some extant an unattainable goal, but we can can get ever closer. This improvement of our knowledge and understanding is one of the driving factors that justify scientific research.

 

[JeffKoch]

The second word in the subject line answers it's own question: There is no such thing as a fictitious force, so obviously the answer is "no".

If you refer to forces that are best described, and are felt and can do work, in non-inertial frames of reference, I don't know if the answer is a definitive "yes" or "no", but it makes sense to use descriptions of forces appropriate to a frame of reference when you are trying to understand what is happening in that frame of reference.

 

[D H]

The second word in the subject line answers it's own question: There is no such thing as a fictitious force, so obviously the answer is "no".

Better said: There are no fictitious forces in an inertial frame.

When the question implicitly or explicitly asks for the forces in a non-inertial frame (e.g., post #3), the answer is "yes". There is no way to answer the question Hootenanny asked in [post=3549926]post #3[/post] without using fictitious forces.

When the question is restricted to predicting outcomes as MikeyW proposed in [post=3549947]post #6[/post], the answer is, in theory, "no" since there are no fictitious forces in an inertial frame. In practice, the answer is still "yes". I gave several examples in [post=3549961]post #8[/post] where nobody in their right mind would even begin trying to answer the question from the perspective of an inertial frame.

 

[harrylin]

[..]
Then my term 'perceived acceleration' is simply your 'co-ordinate acceleration'.
[..]
Problem is, it is the wrong use of Newton's 2nd law. You cannot claim to use the 2nd law (acceleration is proportional to force) if the whole framework of co-ordinates is, itself, rotating. I can hear my maths master saying it now; 'WRONG! You HAVE TO write the equation of motion.'

Newton's 2nd law specifically refers to the notion of a linear function "Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur". (I don't think we need to speak latin to figure out the meaning of this!)

Right - it's an improper use of Newton's laws, misapplying them with respect to rotating frames.

However, the point of this thread is to put the claim to the test that this improper utilisation is very useful, or even necessary to solve certain problems. And regretfully, so far not a single calculation example with fictitious forces of such a case has been given... Thus the challenge remains open to those who made such claims.

 

[harrylin]

[..] I gave several examples in [post=3549961]post #8[/post] where nobody in their right mind would even begin trying to answer the question from the perspective of an inertial frame.

I doubt that that would be necessary in order to avoid using fictitious forces; but that's what is to be seen! So, instead of continuing with making assertions that cannot be tested, please provide just one detailed calculation example using fictitious force that we can put to the test - that's the purpose of this thread.

 

[D H]

Right - it's an improper use of Newton's laws, misapplying them with respect to rotating frames.

There is no misapplying here. All frames of reference are equally valid. You just have to do the math right.

However, the point of this thread is to put the claim to the test that this improper utilisation is very useful, or even necessary to solve certain problems. And regretfully, so far not a single calculation example with fictitious forces of such a case has been given... Thus the challenge remains open to those who made such claims.

Oh, please. I gave several examples. All you have to do is google those terms. You will find web pages, journal articles, even books on the cited subject. You want a simple example. Such a simple example doesn't exist. If it was just one simple equation there would be no reason to add the complexity of fictitious forces. Fictitious forces vanish in an inertial frame. We add that complexity because there is a whole lot more than one simple equation is involved in those applications.

This is very basic sophomore/junior level physics. It is downright silly to be arguing about it.

 

[harrylin]

There is no misapplying here. All frames of reference are equally valid. You just have to do the math right.

Same as when you apply the Lorentz transformations to accelerating frames: that is misapplication by definition. And of course you can always improvise to make it kind of work outside of the specs (as you say, "just ... do the math right").

Oh, please. I gave several examples. All you have to do is google those terms. You will find web pages, journal articles, even books on the cited subject. You want a simple example. Such a simple example doesn't exist. [..] This is very basic sophomore/junior level physics. It is downright silly to be arguing about it.

Again: this has nothing to do with basic level physic. The purpose here is to avoid arguing with words (that's useless!) and stick to calculations instead. If you know of no simple calculation example, then please provide a link to a complicated one that you think makes your point.

Harald