Understanding Reduced Mass and its Role in Classical Mechanics

[andrewr]

In classical mechanics, reduced mass is a way to conveniently take a two body problem and reduce it to a one body problem with an artificial "origin". It simplifies calculations for orbits and angular momentum, etc.

I am familiar with the lorentz transform and the γ factor, but am not proficient with four vectors & tensors, etc.

I have been told that the equivalent concept to a "center of mass" is a barycentric system where the total momentum is zero. I assume that's directed momentum, as clearly a rotating system has angular momentum which is non-zero.

I was thinking that in the case of a set of two masses at rest; should they be accelerated translationally as opposed to rotationally, the problem of how energy must apportion itself among these two masses is fixed by the fact that they have the same linear velocity if the system is to maintain a non-spreading (on average) of the distance between masses.

So, given m1, and m2, a single value gamma, will describe how each mass's total value will be when a quantity of energy is added.

m1*(γ -1) + m2*(γ-1) = (m1+m2)*(γ-1) = K.E./c**2
Therefore if E energy is added to a system at rest (purely non-torque wise); the amount of energy and therefore additional mass that m1 and m2 gain are fixed by their rest masses.

In the barycentric system, however, I am not able to see how this would work out for sure: eg if a purely torque causing energy 'E' were added to a system of two masses at rest, what would be the increase in mass of each of the two masses that are now in rotational motion of energy E.

In the linear translation, the coupling between the masses wasn't important -- but in rotational motion there has to be something which maintains the acceleration of the individual masses so that they don't fly apart (radius between them becoming non-fixed on average; though a change in r with E is possible otherwise.).

My questions, then, are the following two::
Does it matter how they are held together (eg: mass-less gedanken chain, gravity, electrostatic attraction) in determining how the energy will distribute itself among the two masses?

And am I correct in thinking that total momentum equalling zero would mean that, for the masses going instantaneously in opposite directions x, but separated by distance y with no relative y motion;
during the one instant per revolution that this condition holds, it is the momentum condition
$$\frac{m_{1}v_{1}}{\sqrt{1-(v_{1}/c)^{2}}}=\frac{m_{2}v_{2}}{\sqrt{1-(v_{2}/c)^{2}}}$$
; or does it amount to some other relationship?

 

[Bill_K]

And am I correct in thinking that total momentum equalling zero would mean that, for the masses going instantaneously in opposite directions x, but separated by distance y with no relative y motion; during the one instant per revolution that this condition holds, it is the momentum condition
$$\frac{m_{1}v_{1}}{\sqrt{1-(v_{1}/c)^{2}}}=\frac{m_{2}v_{2}}{\sqrt{1-(v_{2}/c)^{2}}}$$

Yes. This is the answer to all your questions! But my suggestion, andrewr, is to steer away from concepts that, while being most familiar in Newtonian mechanics, are not as useful in special relativity. Kinetic energy is among these.

 

[andrewr]

Yes. This is the answer to all your questions! But my suggestion, andrewr, is to steer away from concepts that, while being most familiar in Newtonian mechanics, are not as useful in special relativity. Kinetic energy is among these.

Hi Bill, You sure answer a lot of questions! :)
Thanks! Humor me a bit more, in my gedanken here...

I am needing to relate special relativity to classical mechanics because Schrodinger's is based on classical formulations of energy where mass stays constant. Kinetic energy is equivalent to gained mass, and so that is what I am interested in pursuing. I don't care if it is called kinetic energy or mass...

What I am interested in doing in this question is finding out a general way / correction to reduced mass problems that takes into account relativistic effects. It doesn't need to be exact, but qualitatively correct.

Taking your affirmation, and working with my assumptions, I get the following as the relationship between velocities for two masses that are in pure rotational motion:

$$v_{2}=\frac{m_{1}}{m_{2}}\left(\frac{1}{v_{1}^{2}}+\frac{\left(\frac{m_{1}}{m_{2}}\right)^{2}-1}{c^{2}}\right)^{-\frac{1}{2}}$$

At low velocities it reduces to the classical solution: $$v_{2}\sim=\frac{m_{1}}{m_{2}}v_{1}.$$
The classical solution has a central point, the "center of mass" point which is at a fixed distance from each mass -- and effectively makes a fixed "origin" around which these masses circulate.

I *think* (correct me if I am wrong) if the system of two masses is in pure rotational motion the angular frequency of their orbit (radians/second) has to be the same.

If that is correct, then the ratio of the radii to a fixed/motionless origin (in the relativistic system) must be directly proportional to the ratio of the two velocities of these point masses and I can claim:

$$\frac{r_{2}}{r_{1}}=\frac{v_{2}}{v_{1}}=\frac{m_{1}}{m_{2}}\left(1+\frac{v_{1}^{2}}{c^{2}}\left(\left(\frac{m_{1}}{m_{2}}\right)^{2}-1\right)\right)^{-\frac{1}{2}}$$
so, for example, if m1=10*m2,
$$\frac{r_{2}}{r_{1}}=\frac{1}{\sqrt{.01+\frac{v_{1}^{2}}{c^{2}}.99}}$$

Does this look correct? So if I added mass in the form of energy to the system, it would distribute itself among the two masses in inverse proportion to the rest mass they each have; that means the smaller mass will increase faster than the bigger one, and that will act as negative feedback so that the lesser effective mass adsorbs successively less proportion of the energy until the effective masses of m1 & m2 are equal -- at which point no more energy could be added, since the velocity would be 'c'.

So, the bottom line is that the ratio of radii of all systems tends toward 1:1 as total energy is increased.
Now that I have an intuitive idea of how the system will deform; what does "barymetric" mean in terms of my example?

Would GR have a different result than the one I have come up with due to the accelerations; or does the analysis stay the same... (Eg: might there be more correction needed if GR is right over the correction needed if only SR is right?)

Thanks. :)

 

[Meir Achuz]

What you trying cannot be done my manipulating masses.
You must use the barycentric system for two bodies.

 

[andrewr]

What you trying cannot be done my manipulating masses.
You must use the barycentric system for two bodies.

I don't know what a barycentric system is. What is fundamentally wrong with adding energy to mass such that the relativistic momentum is balanced around a point, but with radii varying with total energy added?
What is the crux of the error I am going to encounter by approaching it the way I am?

 

[Meir Achuz]

I have been told that the equivalent concept to a "center of mass" is a barycentric system where the total momentum is zero.

And am I correct in thinking that total momentum equalling zero would mean that, for the masses going instantaneously in opposite directions x, but separated by distance y with no relative y motion;
during the one instant per revolution that this condition holds, it is the momentum condition
$$\frac{m_{1}v_{1}}{\sqrt{1-(v_{1}/c)^{2}}}=\frac{m_{2}v_{2}}{\sqrt{1-(v_{2}/c)^{2}}}$$

These two statements are correct and show that you do know what a barycentric system is. All your other statements and equations are wrong. Start over in the barycentric system, using momentum rather than velocity.

 

[andrewr]

These two statements are correct and show that you do know what a barycentric system is. All your other statements and equations are wrong. Start over in the barycentric system, using momentum rather than velocity.

Thank you for giving me a working definition of barycentric. I at least now understand what the word is connected with.

I derived the first equation with velocity from the second statement regarding momentum which includes velocity 1 and velocity 2. So I presume I must have made a mistake in the derivation, and am not sure what you mean about "it can't be done" in terms of velocity. This is what I did, to simplify the math -- assume velocity is in units of 'c=1';

$$\frac{m_{1}v_{1}}{\sqrt{1-v_{1}^{2}}}=\frac{m_{2}v_{2}}{\sqrt{1-v_{2}^{2}}} \Longrightarrow \frac{m_{1}}{\sqrt{v_{1}^{-2}-1}}=\frac{m_{2}}{\sqrt{v_{2}^{-2}-1}}$$

If that is not wrong, then neither is this:

$$\frac{v_{2}^{-2}-1}{v_{1}^{-2}-1}=\left(\frac{m_{2}}{m_{1}}\right)^{2}$$

nor this:

$$\frac{v_{2}^{-2}-1}{v_{1}^{-2}-1}=\left(\frac{m_{2}}{m_{1}}\right)^{2}\Longrightarrow v_{2}^{-2}=\left(\frac{m_{2}}{m_{1}}\right)^{2}v_{1}^{-2}-\left(\frac{m_{2}}{m_{1}}\right)^{2}+1$$

and lastly:

$$v_{2}=\left(\left(\frac{m_{2}}{m_{1}}\right)^{2}v_{1}^{-2}+1-\left(\frac{m_{2}}{m_{1}}\right)^{2}\right)^{-\frac{1}{2}}$$

And in the limit of v1->0, dividing by v1**-2 will cause that term to dominate, and as one gets closer to zero velocity the following becomes true:

$$v_{2}=\left(\left(\frac{m_{2}}{m_{1}}\right)^{2}v_{1}^{-2}+1-\left(\frac{m_{2}}{m_{1}}\right)^{2}\right)^{-\frac{1}{2}}\sim=lim(v_{1}\rightarrow0):\left(\left(\frac{m_{2}}{m_{1}}\right)^{2}v_{1}^{-2}\right)^{-\frac{1}{2}}=\frac{m_{1}}{m_{2}}v1$$

Which is exactly what I derived before:

$$v_{2}\sim=\frac{m_{1}}{m_{2}}v_{1}.$$

Since you say "all your other ... are wrong", which included the above statement -- what do you mean?
It's just a mathematical derivation from the first premise as far as I can see.

 

[andrewr]

Okay, no reply. Does anyone else have any idea what Meir Achuz might have meant?

In the second leg of my study, which is wrong according to Meir Achuz, I simply noted that angular frequency needs to be fixed regardless of radius or else the masses in orbit about the origin point would move such that they are not diametrically opposed at all times. That would mean that at some point in time, that the radii for each mass would be in the same direction rather than opposite -- and therefore the total momentum would add rather than subtract -- which violates the system being barycentric (zero total momentum) at all times.

I can accept I made a mistake in that, but I don't know what it would be. The logic seems solid to me, still. Does anyone have a clear explanation?

The result of my second study is that the ratio of the two radii are affected by the addition of energy of motion to the two masses. That relationship does nothing to say what the total radius of the two radii added would be. So, when I try to compute total energy (or mass) I have a variable that is undefined in the problem, and so it appears there *might be* many possible systems of radii that would have the same total energy. Therefore, my result isn't very useful by itself, but I don't see that it is actually wrong. Does anyone disagree, and can state clearly why? This is just algebra as far as I can see.

There may be different theories (General Relativity, etc) that affect the meaning of "radius", or something to do with acceleration and energy, but it isn't implicit in the problem which I am working out using only special relativity. Again, this isn't a homework problem -- and I am not cheating by asking it, so giving me false data isn't going to get me in trouble with a professor -- it's just going to irritate me and won't make the world a better place by removing a cheater. The length of time that I work on a problem ought to be evidence enough that it isn't homework. I don't follow the pattern that colleges use. I hope I am over-analyzing the response I get on these forums, but I can't explain the weird behavior I encounter here -- and am grasping at straws to try and understand it and overcome it.

Peace.

 

[pervect]

You write

I am needing to relate special relativity to classical mechanics because Schrodinger's is based on classical formulations of energy where mass stays constant. Kinetic energy is equivalent to gained mass, and so that is what I am interested in pursuing. I don't care if it is called kinetic energy or mass...

It sounds like what you are trying to do is to use Newton's law of gravity together with relativistic mechanics -or rather to use a "revised form of Newton's law, where by you change the Newtonian mass m to the total energy / c^2.

The problem with this is that the basic idea is not right, even if you carry out the calculations correct.y. The biggest issue is that the force you get from Newton's force law does NOT transform as a 4-vector. Something that you can (hopefully!) check for yourself if you take the time to do so - at least you sound like you have the background to be able to do that.

So you're mixing together apples (things that transform as 4-vectors) and banana's (things that don't transform as 4 vectors), and unfortunately that leads, metaphorically, to painful stomach conditions. Non-metaphorically, it leads to your results not having much physical significance, as the answer you get will be frame dependent - you can only achieve frame independence, also known as "covariance", if all the quantities that you use in yoru calculation transform in a covariant manner.

To expand on this point a bit - Coulomb's force law for electrostatics isn't covariant in isolation either. Electromagnetism as a whole is covariant, but you need to include both the electric and magnetic forces in an analysis to make it that way. Just snipping out the Coulomb force law and taking it in isolation, it's not covariant.

Therefore, you won't expect whatever calculations you do to be covariant, i.e. the answers you get will depend on your frame of reference.

It's not quite clear what experimental situation you are trying to analyze, but the lack of covariance in your approach is what I think most are objecting to.

To get covariant results, you'd need to include at a minimum gravitomagnetism - which is still a "low velocity" approximation to GR. See for instance http://en.wikipedia.org/w/index.php?title=Gravitomagnetism&oldid=424896586

Depending on exactly what you want to accomplish, gravitomagnetism may or may not be a good enough approximation.

 

[Meir Achuz]

$$\frac{m_{1}v_{1}}{\sqrt{1-v_{1}^{2}}}=\frac{m_{2}v_{2}}{\sqrt{1-v_{2}^{2}}} \Longrightarrow \frac{m_{1}}{\sqrt{v_{1}^{-2}-1}}=\frac{m_{2}}{\sqrt{v_{2}^{-2}-1}}$$
If that is not wrong, then neither is this:
$$\frac{v_{2}^{-2}-1}{v_{1}^{-2}-1}=\left(\frac{m_{2}}{m_{1}}\right)^{2}$$

Since you say "all your other ... are wrong", which included the above statement -- what do you mean?
It's just a mathematical derivation from the first premise as far as I can see.

Prevect gave a good answer, but I should clarify what I meant by 'wrong'. Your premises and interpretations are wrong even if the algebra is correct.

"what would be the increase in mass of each of the two masses that are now in rotational motion of energy E." is a meaningless question, since their individual masses do not increase.
"m1*(γ -1) + m2*(γ-1) = (m1+m2)*(γ-1) = K.E./c**2
Therefore if E energy is added to a system at rest (purely non-torque wise); the amount of energy and therefore additional mass that m1 and m2 gain are fixed by their rest masses." is all wrong.

Some other passages like
"So if I added mass in the form of energy to the system, it would distribute itself among the two masses in inverse proportion to the rest mass they each have; that means the smaller mass will increase faster than the bigger one, and that will act as negative feedback so that the lesser effective mass adsorbs successively less proportion of the energy until the effective masses of m1 & m2 are equal -- at which point no more energy could be added, since the velocity would be 'c'."
make so little sense that they 'are not even wrong'.

 

[andrewr]

You write
It sounds like what you are trying to do is to use Newton's law of gravity together with relativistic mechanics -or rather to use a "revised form of Newton's law, where by you change the Newtonian mass m to the total energy / c^2.

Yes, though I wasn't interested in gravity because atomic physics is hardly affected by that small of a force. I was just concerned that there may be something in General Realtivity that would seriously affect a two body problem rotating at relativistic speeds and that the pitfall (and its name, not just "wrong") might be obvious to someone familiar with it.

The problem with this is that the basic idea is not right, even if you carry out the calculations correct.y. The biggest issue is that the force you get from Newton's force law does NOT transform as a 4-vector. Something that you can (hopefully!) check for yourself if you take the time to do so - at least you sound like you have the background to be able to do that.

I didn't realize I was using Newtonian force law. That's why I originally asked if it made a difference whether this is a rigid body (mass-less rod connecting two dissimilar weights) or if this approach would work in a force system as well. Are you saying it would not work in a rigid body problem either?

It's strange, because I have derived that the units of a angular momentum are [J.s] from what I can tell in relativity applied the way I am using it -- which is identical to Plank's constant -- and the square root of 3/4 shows up the particular form I am using it -- identical to the equation for spin of an electron. It pops out as a natural consequence of the approximate corrections I am making based on the ideas in the thread.

There are several other results which were very encouraging as well -- but this is going to get me in trouble at some point is what I am hearing.

So you're mixing together apples (things that transform as 4-vectors) and banana's (things that don't transform as 4 vectors), and unfortunately that leads, metaphorically, to painful stomach conditions. Non-metaphorically, it leads to your results not having much physical significance, as the answer you get will be frame dependent - you can only achieve frame independence, also known as "covariance", if all the quantities that you use in yoru calculation transform in a covariant manner.

I have a particular frame that this needs to be true in -- the point that these two masses would naturally rotate around as an inertial frame. I am not trying to be frame independent.. Does that still cause a problem? All I am interested to know is if the energy of the system (total energy) is to be changed, that in the lab frame of the centroid (a non accelerating/rotating frame) that any energy traded between the lab frame and the two masses would balance. How the energy gets there isn't of much concern -- that's a more advanced problem. But I do need to know the relative distance from the centroid to each of the masses using rest frame measurements. That's all I need.

Is there perhaps a similar problem/example that is worked out to some kind of useful answer -- that I could read? It seems like learning a large amount of GR formalism would take far longer than just asking the question and guidance.

To expand on this point a bit - Coulomb's force law for electrostatics isn't covariant in isolation either. Electromagnetism as a whole is covariant, but you need to include both the electric and magnetic forces in an analysis to make it that way. Just snipping out the Coulomb force law and taking it in isolation, it's not covariant.

Ok, that's helpful. I am being forced to use three variables to represent the Coulomb field as it is. The time delay propagation is what I figured was a good approximation to the magnetic field. It seems to have the right magnitude and direction characteristics. In this thread I didn't bother, because I was thinking of charge neutral masses. It still makes a difference?

Therefore, you won't expect whatever calculations you do to be covariant, i.e. the answers you get will depend on your frame of reference.

It's not quite clear what experimental situation you are trying to analyze, but the lack of covariance in your approach is what I think most are objecting to.

I appreciate your explanation of what other's are probably seeing -- it is just confusing to me.

My apologies for not being very clear but not knowing the GR vocabulary is part of the problem.

To get covariant results, you'd need to include at a minimum gravitomagnetism - which is still a "low velocity" approximation to GR. See for instance http://en.wikipedia.org/w/index.php?title=Gravitomagnetism&oldid=424896586

Depending on exactly what you want to accomplish, gravitomagnetism may or may not be a good enough approximation.

OK, I'll look into that; all I am doing is a first order type correction to the Hammiltonian of Schrodinger's equation re-written in a computer solvable form. I don't need exact answers, but I do need the different effects from relativity to show up qualitatively -- (gravity is pretty small in the classical case) but that includes fine constant structure. Dirac's equations and other peterbation theories are both too cumbersome for me to apply to a multi-body system, and don't even get qualitatively correct results with respect to fine structure in Hydrogen (supposedly an exact case). I have just barely grasped the problem with "spin" vs. "angular orbital momentum" and am beginning to get results that are promising -- but I am aware it may dead end abruptly.

I appreciate your explanations.