Are fictitious forces necessary to solve certain problems?

[A.T.]

And of course you can always improvise to make it kind of work

What do you mean by "improvise"? The point of having the concept of inertial forces is that you don't have to improvise, and derive ad-hoc corrections for each non-inertial frame case. Instead you apply a consistent set of rules to get the correct results. And it doesn't just "kind of work". It works perfectly. It is not more of an "improvisation" than Newton's unmodified laws.

 

[Dale]

Develop the equations of motion for a satellite in a Lissajous orbit about the sun-earth L1 point.

If you know of no simple calculation example, then please provide a link to a complicated one that you think makes your point.

http://www.cds.caltech.edu/archive/help/uploads/wiki/files/39/lecture_halo_2004.pdf

 

[harrylin]

http://www.cds.caltech.edu/archive/help/uploads/wiki/files/39/lecture_halo_2004.pdf

Thanks! But did you give this to demonstrate that fictitious forces are not necessary? Because no such forces are used there, as far as I can see.

 

[D H]

Thanks! But did you give this to demonstrate that fictitious forces are not necessary?

Nobody has said that fictitious forces are necessary. It is just that in some cases the problem becomes intractable or overly complex without the use of such devices.

Because no such forces are used there, as far as I can see.

Sure they are. Slide #2 starts with "Recall equations of CR3BP" and then presents the equations of motion for this system. Those equations of motion are those of a spacecraft (labeled S/C in the figure) expressed in a rotating frame of reference.

Some background: CR3BP (some use CRTBP) is short for "circular restricted 3 body problem" (or "circular restricted three body problem" in the case of CRTBP). I strongly suggest you google those two phrases.

The subject of the CR3BP is the motion of a very, very small third body in the presence of a pair of bodies in circular orbits about their center of mass. (The more general problem of the elliptical restricted 3 body problem is a much tougher nut to crack.) Of the two massive bodies, one will be more massive than the other. This larger body is called the primary body, the smaller one, the secondary body. Restricting the third body to having a mass that is many, many orders of magnitude smaller than that of the secondary means that the effect of the third body on the behaviors of the primary and secondary bodies will be negligible and can be ignored.

Those equations of motion are not expressed in SI units. They are instead expressed in units such that

• One mass unit is the sum of the masses of the primary and secondary bodies. In these units, the secondary body has mass μ; the primary body has mass 1-μ. The primary is by definition the more massive of the two bodies, meaning that μ is between 0 and 1/2.
• One distance unit is the distance between the primary and secondary bodies. This distance is constant since the primary and secondary a two bodies are in circular orbits about one another.
• One time unit is the orbital period the primary and secondary bodies divided by (2*pi).

Note that, by definition, this system of units yields numeric values of one for the total mass of the system and for the orbital radius. A couple of other key quantities also have a numeric value of one in this system of units. These are the universal gravitational constant G and the magnitude of the primary and secondary's angular velocity vector ω.

Working in inertial coordinates would yield nine coupled, non-linear second order differential equations: An absolute mess. Switching to a frame that is rotating with the orbit of the primary and secondary about their center of mass simplifies things immensely. The primary and secondary are not moving in this frame. Six of those nine coupled, non-linear second order differential equations just vanish. The three equations of motion that remain describe the body of interest, the third body. Those three equations of motion now include terms due to the fictitious centrifugal acceleration, but this is a very small price to pay for having six of the original equations of motion just vanish.

 

[harrylin]

[..] Sure [fictitious forces] are [used there]. Slide #2 starts with "Recall equations of CR3BP" and then presents the equations of motion for this system. Those equations of motion are those of a spacecraft (labeled S/C in the figure) expressed in a rotating frame of reference.

Some background: CR3BP (some use CRTBP) is short for "circular restricted 3 body problem" (or "circular restricted three body problem" in the case of CRTBP). I strongly suggest you google those two phrases.

OK - so you claim that fictitious forces were used to derive those equations. Slowly we are getting somewhere.

Now we only have to find a presentation of such a derivation with fictitious forces, and which we can then compare with the equivalent derivation without such forces, if it's practically doable.

So, checked with Google and found for example:
http://www.cdeagle.com/ommatlab/crtbp.pdf
However, again I noticed no reference to fictitious forces!

The subject of the CR3BP is the motion of a very, very small third body in the presence of a pair of bodies in circular orbits about their center of mass. (The more general problem of the elliptical restricted 3 body problem is a much tougher nut to crack.) Of the two massive bodies, one will be more massive than the other. This larger body is called the primary body, the smaller one, the secondary body. Restricting the third body to having a mass that is many, many orders of magnitude smaller than that of the secondary means that the effect of the third body on the behaviors of the primary and secondary bodies will be negligible and can be ignored.

Those equations of motion are not expressed in SI units. They are instead expressed in units such that

• One mass unit is the sum of the masses of the primary and secondary bodies. In these units, the secondary body has mass μ; the primary body has mass 1-μ. The primary is by definition the more massive of the two bodies, meaning that μ is between 0 and 1/2.
• One distance unit is the distance between the primary and secondary bodies. This distance is constant since the primary and secondary a two bodies are in circular orbits about one another.
• One time unit is the orbital period the primary and secondary bodies divided by (2*pi).

Note that, by definition, this system of units yields numeric values of one for the total mass of the system and for the orbital radius. A couple of other key quantities also have a numeric value of one in this system of units. These are the universal gravitational constant G and the magnitude of the primary and secondary's angular velocity vector ω.

Thanks for the clarification.

Working in inertial coordinates would yield nine coupled, non-linear second order differential equations: An absolute mess. Switching to a frame that is rotating with the orbit of the primary and secondary about their center of mass simplifies things immensely.

I fully agree; and that was never an issue. It's a common misconception to think that one has to use fictitious forces in order to map equations of motion to a rotating frame.

The primary and secondary are not moving in this frame. Six of those nine coupled, non-linear second order differential equations just vanish. The three equations of motion that remain describe the body of interest, the third body. Those three equations of motion now include terms due to the fictitious centrifugal acceleration, but this is a very small price to pay for having six of the original equations of motion just vanish.

Coordinate acceleration should not be confounded with fictitious force - those are unrelated concepts. And I did not spot a fictitious force in the derivation above.Harald

 

[D H]

Coordinate acceleration should not be confounded with fictitious force - those are unrelated concepts. And I did not spot a fictitious force in the derivation above.

There you go then. This is the source of your confusion. In Newtonian mechanics, coordinate acceleration and fictitious forces are essentially same thing, sans a factor of mass. The net fictitious force is simply coordinate acceleration times mass.

 

[Dale]

It's a common misconception to think that one has to use fictitious forces in order to map equations of motion to a rotating frame.

This is not correct. One does have to use ficititious forces in a rotating frame, otherwise the equations of motion are incorrect.

What one does not have to do is to stick a big label on them and say "this term here is a fictitious force". The appropriate terms in the equations of motion represent fictitious forces whether or not they are explicitly labeled as such.

 

[Dale]

In Newtonian mechanics, coordinate acceleration and fictitious forces are essentially same thing, sans a factor of mass.

To further emphasize this point, fictitious forces are always proportional to mass, so you can always drop or add a factor of mass to go between the two.

 

[harrylin]

There you go then. This is the source of your confusion. In Newtonian mechanics, coordinate acceleration and fictitious forces are essentially same thing, sans a factor of mass. The net fictitious force is simply coordinate acceleration times mass.

You mean your confusion.
But indeed, this seems to be largely a matter of words! In Newtonian mechanics as well as most textbooks (including the one that you directed me to by means of Google), coordinate acceleration exists due to Newtonian ("real") forces, and no fictitious force concepts are introduced at all.

[..]What one does not have to do is to stick a big label on them and say "this term here is a fictitious force". The appropriate terms in the equations of motion represent fictitious forces whether or not they are explicitly labeled as such.

What you call "fictitious force", others might call an artifact or correction term for non-inertial motion; and although mathematically the value will be the same, conceptually that is very different. So, it's not merely a matter of labels, but also of concepts. Perhaps that is why some teachers can get very upset when others call those correction terms "fictitious forces".

Anyway, as commonly textbooks do not use the fictitious force concept for those derivations, I take it that my question has been sufficiently answered.

Thanks for the feedback!

 

[A.T.]

coordinate acceleration exists due to Newtonian ("real") forces, and no fictitious force concepts are introduced at all.

In general coordinate acceleration depends on the net force, which is the sum of all forces that might act: interaction and inertial.

 

[Dale]

What you call "fictitious force", others might call an artifact or correction term for non-inertial motion; and although mathematically the value will be the same, conceptually that is very different. So, it's not merely a matter of labels, but also of concepts. Perhaps that is why some teachers can get very upset when others call those correction terms "fictitious forces".

It doesn't matter if you also call it an "artifact" or a "correction term" or "Bob's uncle". It fits the definition of a fictitious force therefore it is a fictitious force, regardless of what other definitions it also fits.

Your argument here is like saying that a square is not a rectangle because other people will call it a square, and it isn't just a matter of labels since squares and rectangles are conceptually different, and some people get upset if you call a square a rectangle. It is an invalid argument. A square is a rectangle because it fits the definition of a rectangle, and the extra terms in the equations of motion in a non-inertial frame are fictitious forces because they fit the definition of a fictitious force.

Anyway, as commonly textbooks do not use the fictitious force concept for those derivations, I take it that my question has been sufficiently answered.

Kindly back up this claim with a reference. All textbooks should use fictitious forces, either as a part of the derivation or as an end result of the derivation. If they do not, then they are in error. Obviously, they may not discuss their use of fictitious forces, but they must use them.

 

[harrylin]

[..]
Kindly back up this claim with a reference. All textbooks should use fictitious forces, either as a part of the derivation or as an end result of the derivation. If they do not, then they are in error. Obviously, they may not discuss their use of fictitious forces, but they must use them.

Already given and commented in post #65; similar to basic textbooks that discuss Coriolis acceleration etc. without introducing the fictitious force concept. It was in that sense that I intended my question, which now has been answered to my satisfaction.

 

[D H]

Already given and commented in post #65; similar to basic textbooks that discuss Coriolis acceleration etc. without introducing the fictitious force concept. It was in that sense that I intended my question, which now has been answered to my satisfaction.

Coriolis acceleration and coriolis force are one and the same thing, sans a factor of mass. You are playing a stupid semantics game, Harald.

Thread closed.