Are Fictituous Forces Conservative?

[nonequilibrium]

Hello,

I was wondering: are fictituous forces conservative?

The reason I ask: if I want to describe two gravitationally interacting particles, can I take the non-inertial frame of reference with the origin in the center of mass and rotating along with the masses such that from inside this reference frame the bodies don't seem to rotate, and making Newton's laws valid (and thus also Lagrange's) by introducing the appropriate pseudo-forces?

Thank you!

 

[dgOnPhys]

Hello,

I was wondering: are fictituous forces conservative?

The reason I ask: if I want to describe two gravitationally interacting particles, can I take the non-inertial frame of reference with the origin in the center of mass and rotating along with the masses such that from inside this reference frame the bodies don't seem to rotate, and making Newton's laws valid (and thus also Lagrange's) by introducing the appropriate pseudo-forces?

Thank you!

Hello vodka,

interesting question, from what I gathered I would say that only the centrifugal force is strictly conservative (can be expressed as a gradient), neither Coriolis nor Euler fictitious forces appear to be conservative: the first depends on velocity, the second does not create a closed differential form.
Still you might be able to write them using generalized potentials in a Lagrangian.

In your specific case, in general your reference will be subject to all 3 types of fictitious forces so writing conservation of energy equations won't be straightforward.

Of course you can still write Newton's equations, you do not need conservative forces for that

I guess you can settle with the usual approaches:
- conservation of angular momentum limits motion to a plane
- reduced mass let's you write the equation with the origin on one of the masses

Hope this helps

 

[K^2]

There could be a whole bunch of conservative fictitious forces. Simplest case, just take a uniformly accelerating reference frame. But yes, in general, it doesn't have to be conservative. Coriolis force is an excellent example.

 

[tiny-tim]

hello mr. vodka!

fictitious forces do no work, and therefore must be conservative

the https://www.physicsforums.com/library.php?do=view_item&itemid=84"

and the https://www.physicsforums.com/library.php?do=view_item&itemid=86" obviously does no work (because … like magnetism … it is perpendicular to the velocity, ie the velocity in the rotating frame)

this is not surprising … from the work-energy theorem, if there is no https://www.physicsforums.com/library.php?do=view_item&itemid=75" since the fictitious forces are specifically designed to enable Newton's law to apply!

(i find it difficult to see how the concept would apply in relation to the Euler force, since that only applies in a non-uniformly rotating frame … but in principle, that should also be conservative, and for the same reason)

 

[dgOnPhys]

hello mr. vodka!

fictitious forces do no work, and therefore must be conservative

the https://www.physicsforums.com/library.php?do=view_item&itemid=84"

and the https://www.physicsforums.com/library.php?do=view_item&itemid=86" obviously does no work (because … like magnetism … it is perpendicular to the velocity, ie the velocity in the rotating frame)

this is not surprising … from the work-energy theorem, if there is no https://www.physicsforums.com/library.php?do=view_item&itemid=75" since the fictitious forces are specifically designed to enable Newton's law to apply!

(i find it difficult to see how the concept would apply in relation to the Euler force, since that only applies in a non-uniformly rotating frame … but in principle, that should also be conservative, and for the same reason)

Hi tiny-tim,

I am not really following your line of reasoning:
- Euler force cannot be expressed as a gradient (F.ds is not a closed differential form) so it cannot be conservative. Even in the simple OP case, the CM frame rotating with the masses will experience angular acceleration (e.g. an elliptical planetary motion) and this force will perform work
- not all forces that depend only on position are conservative
- centrifugal force is conservative, its potential easy to write (not just in the OP case) and it does do work in general
- Coriolis force -as you point out- does no work so I guess we do not have to bother with its potential (even though I am not sure how to write this force as a gradient...)
- what do you mean by "the fictitious forces do no work in the inertial frame"?
- work is not frame invariant so how are you using work-energy theorem?

sorry for the bashing but this is the first time I see a post of yours that is not right on spot... could not resist the chance ;)

 

[tiny-tim]

hi dgOnPhys!

- not all forces that depend only on position are conservative

yes, you're right … i was thinking of distance when i wrote that …

i should have said "it depends only on distance (from the centre), and so it is obviously a conservative force

- what do you mean by "the fictitious forces do no work in the inertial frame"?
- work is not frame invariant so how are you using work-energy theorem?

yes, work is not frame invariant

the odd thing is that, in a uniformly rotating frame, the ficititious forces are conservative, so since the work-energy theorem applies in both frames, there should be a straightforward way of using that theorem to prove it (instead of looking at the definition of the centrifugal and Corliois forces) …

but i can't see one yet … i'll have to think about it some more