2-body problem - conservation of angular momentum

In summary: If you consider extended bodies then angular momentum is conserved because the angular momentum of the whole is the sum of the angular momenta of the parts. The implicit assumption is that the interactions between extended bodies are accurately described as a simple linear superposition of the forces between all of the pairs of interacting parts.
  • #1
dyn
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Hi
With the 2-body problem relating to planetary orbits i have encountered the following ; the gravitational force on the reduced mass acts towards the large mass(Sun) and since it is a central force it exerts no torque about the fixed centre(Sun) so angular momentum is conserved.
Conservation of angular momentum states that the total angular momentum of an isolated system is always conserved ; so here's my question - the 2-body problem (Sun and orbiting planet) is an isolated system so can i just that its angular momentum must be conserved straight away without having to consider that there is no torque about its centre due to force being central ?
Thanks
 
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  • #2
Sure. The angular momentum of any system that is isolated, in the sense that no net external torque acts on it, is conserved.
 
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  • #3
Thank you. So if I have a 2-body problem and consider that as an isolated system its angular momentum is constant regardless of the force between the 2 bodies ? Even if that force is non-conservative or not a central force ?
 
  • #4
Yes.
 
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  • #5
The derivation goes as follows. First start with the equations of motion for the two particles,
$$m_1 \ddot{\vec{r}}_1 =-\frac{G m_1 m_2}{|\vec{r}_1-\vec{r}_2|^3} (\vec{r}_1-\vec{r}_2),$$
$$m_2 \ddot{\vec{r}}_2 =+\frac{G m_1 m_2}{|\vec{r}_1-\vec{r}_2|^3} (\vec{r}_1-\vec{r}_2).$$
Now introduce center-mass and relative coordinates
$$\vec{R}=\frac{1}{M} (m_1 \vec{r}_1+ m_2 \vec{r}_2), \quad \vec{r}=\vec{r}_1-\vec{r}_2,$$
where ##M=m_1+m_2##.

Now note that by adding of the equations of motion you get
$$M \ddot{\vec{R}} = m_1 \ddot{\vec{r}}_1 + m_2 \ddot{\vec{r}}_2=0,$$
i.e., the center of mass is moving like a free particle uniformly. This is of course nothing else than momentum conservation, i.e., the total momentum ##\vec{P}=m_1 \dot{\vec{r}}_1 + m_2 \dot{\vec{r}}_2=M \dot{\vec{R}}=\text{const}##.

Now express one of the equations of motion in terms of ##\vec{R}##, ##\vec{r}##, and their time derivatives, using this result of total-momentum conservation to get an equation of motion for ##\vec{r}##. The result is [edit: Correction in view of #6]
$$\mu \ddot{\vec{r}}=-\frac{G m_1 m_2 \vec{r}}{r^3}, \quad \mu = \frac{m_1 m_2}{M}.$$
What do you get by taking the cross product of this equation with ##\vec{r}##?
 
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  • #6
vanhees71 said:
Now express one of the equations of motion in terms of ##\vec{R}##, ##\vec{r}##, and their time derivatives, using this result of total-momentum conservation to get an equation of motion for ##\vec{r}##. The result is
$$\mu \ddot{\vec{r}}=-\frac{G m_1 m_2}{r}, \quad \mu = \frac{m_1 m_2}{M}.$$
What do you get by taking the cross product of this equation with ##\vec{r}##?
That equations seems a bit strange to me. On the left hand side is a vector and the right hand side is a scalar.

I think my confusion is over what internal and external forces are. In the 2-body problem of a planet orbiting the sun i consider the gravitational forces between the 2 masses as internal forces. So with no external forces acting , this means no external torques , so i can automatically say that angular momentum is conserved in the 2-body problem. So i don't understand why books say that angular momentum is conserved due to the central nature of the force. As i understand it , the central nature of the force is irrelevant for the conserved angular momentum
 
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  • #7
dyn said:
So i don't understand why books say that angular momentum is conserved due to the central nature of the force.

Angular momentum actually is conserved due to the central nature of the force. Forces have a build in conservation of linear momentum (thanks to Newton's 3rd law) but it is not obvious that they also conserve angular momentum. There must be an additional restriction that is not included into the definition of force. Why shouldn't it be mentioned in a book?
 
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  • #8
DrStupid said:
Angular momentum actually is conserved due to the central nature of the force. Forces have a build in conservation of linear momentum (thanks to Newton's 3rd law) but it is not obvious that they also conserve angular momentum. There must be an additional restriction that is not included into the definition of force. Why shouldn't it be mentioned in a book?
If we consider a force between two point-like objects at a distance from one another, the implicit assumption is that the force on each body acts on that body and acts in a direction toward or away from the other body. Now Newton's third law conserves angular momentum.

If you consider extended bodies then angular momentum is conserved because the angular momentum of the whole is the sum of the angular momenta of the parts. The implicit assumption is that the interactions between extended bodies are accurately described as a simple linear superposition of the forces between all of the pairs of interacting parts.

I do not know why this is not in books. If it was ever pointed out to me, I do not recall it. My vague recollection of that time many years ago is that the notion seemed obvious enough as to need no mention.
 
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  • #9
jbriggs444 said:
If we consider a force between two point-like objects at a distance from one another, the implicit assumption is that the force on each body acts on that body and acts in a direction toward or away from the other body.

What do you mean with "implicit assumption"? It is neither obvious that a force between two point-like objects needs to act in a direction from one towards the other (unless you consider conservation of angular momentum) nor has it been defined by Newton.
 
  • #10
DrStupid said:
What do you mean with "implicit assumption"? It is neither obvious that a force between two point-like objects needs to act in a direction from one towards the other (unless you consider conservation of angular momentum) nor has it be defined by Newton.
In what other direction can it act? Symmetry demands that it must act in such a way.

It is a quite reasonable assumption in my view. Though magnetism with its force-at-right-angles thing does throw a wrench into the works.
 
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  • #11
jbriggs444 said:
In what other direction can it act?

In any other direction.

jbriggs444 said:
Symmetry demands that it must act in such a way.

There was no symmetry mentioned in your statement above. Where does this assumption come from?
 
  • #12
DrStupid said:
In any other direction.
But that does not give you predictive laws of physics.
 
  • #13
jbriggs444 said:
But that does not give you predictive laws of physics.

Why not?
 
  • #14
DrStupid said:
Why not?
Because there is only one well defined direction for two point-like objects at a distance from one another.

The underlying assumption this time is that the laws of physics are invariant with respect to rotation angle. That is the symmetry in question. I had mistakenly thought that was obvious enough as to need no explicit mention.
 
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  • #15
jbriggs444 said:
Because there is only one well defined direction for two point-like objects at a distance from one another.

Even point-like objects may have properties that define other directions (e.g. spin or dipoles).
 
  • #16
DrStupid said:
Even point-like objects may have properties that define other well defined directions (e.g. spin or dipoles).
What point are you trying to argue here?

That the assumption that forces-at-a-distance between interacting pointlike object must act in the direction between the interacting objects is not made? Or that it is erroneous?

I am arguing that the implict assumption is made.
You appear to be arguing that it is erroneous.

The two claims are not in conflict.
 
  • #17
jbriggs444 said:
That the assumption that forces-at-a-distance between interacting pointlike object must act in the direction between the interacting objects is not made?

Yes, I'm not aware of such an assumption and at least Newton's definition of forces allows for any other direction.

jbriggs444 said:
I am arguing that the implict assumption is made.

Is that just your opinion or can you prove it?
 
  • #18
DrStupid said:
Yes, I'm not aware of such an assumption and at least Newton's definition of forces allows for any other direction.
Is that just your opinion or can you prove it?
I can prove it.

I made it.

QED.
 
  • #19
jbriggs444 said:
I made it.

Sorry, I can't follow you. Where did you prove it and how?
 
  • #20
DrStupid said:
Sorry, I can't follow you. Where did you prove it and how?
You do not prove assumptions. You make them. Sometimes you can ground them on other assumptions and thereby prove them correct [at least as correct as those underlying assumptions]. But such is not required. You always land on something unproven eventually.
 
  • #21
In #16 you claimed to be arguing. Now it is just an assumption? I'm really trying to understand you, but I can't.
 
  • #22
DrStupid said:
In #16 you claimed to be arguing. Now it is just an assumption? I'm really trying to understand you, but I can't.
It seemed that you were arguing about the correctness of an assumption that I'd stated rather than arguing about whether it was reasonable.

But I am baffled about what point you are trying to make.
 
  • #23
I am not sure I understand what is being argued here either. OP's original question has been answered in post #2: The angular momentum of any isolated system is conserved. The word "any" includes "regardless of whether the internal forces are central and/or dissipative". The justification for this was provided in some detail by @jbriggs444 in #8 as a direct consequence of the 3rd law,

If two magnetic dipoles on a frictionless surface are released from rest, the spin and orbital angular momentum of each will change so that the total sum of spin plus orbital angular momenta will be zero. Of course this will not be the case if the dipoles are placed in an external magnetic field.
 
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  • #24
jbriggs444 said:
But I am baffled about what point you are trying to make.

My point is, that forces do not conserve angular momentum by default. That must be included into specific force laws and this is worth to mention. Newton not only mentioned it but he even proved that his universal law of gravitation complies with Kepler's Law of Areas. He didn't do something like that for forces in general.

You are arguing that there is an implict assumption that force between point-like objects always act in a direction from one to the other. I do not know, what that means. Are you saing

1. that you assume it (that would hardly be an argumentation but reather an expression of your personal opinion) or

2. that you think that everybody needs to assume it because
2.a it is obvious for some reason other than conservation of angular momentum (that should be explained) or
2.b it has been published somewhere (that would need a reference) and accepted by the scientific community or

3. do you mean something else (that should be further specified)?
 
  • #25
kuruman said:
The justification for this was provided in some detail by @jbriggs444 in #8 as a direct consequence of the 3rd law

It is not a direct consequence of the 3rd law but a consequence of the 3rd law and an additional implicit assumption. I am missing the justification for this assumption (other than conservation of angular momentum).
 
  • #26
DrStupid said:
It is not a direct consequence of the 3rd law but a consequence of the 3rd law and an additional implicit assumption. I am missing the justification for this assumption (other than conservation of angular momentum).
If you like, it follows from an assumptions that pointlike object are pointlike -- that they have no invisible arrows identifying a preferred orientation, that the force between two pointlike bodies is purely a function of the relative positions of the two bodies and their internal properties, and that the net force between two extended bodies is the sum of the forces between their component pieces, and that the laws of physics do not have a preferred direction. That last one via Noether's theorem ends up being an important one, of course and obviates the rest. But perhaps we should not be invoking Noether's theorem in an I level thread.

All of these assumptions are reasonable enough prior to the discovery of electromagnetism, the attribution of energy and momentum to a field and the elimination of instantaneous action at a distance. Not necessarily complete and correct, but at least reasonable.

To explicitly answer your question in #26:
DrStupid said:
1. that you assume it (that would hardly be an argumentation but reather an expression of your personal opinion) or

2. that you think that everybody needs to assume it because
2.a it is obvious for some reason other than conservation of angular momentum (that should be explained)
If I am working in the framework of classical mechanics and electrostatics, I do assume it (1) and I expect pretty much everyone else to assume it (2) because it is fairly obvious (2a) as above and because the model works within its domain of applicability.

None of this makes the resulting model correct outside that domain, of course. Instantaneous action at a distance turns out to be physically unrealistic.
 
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  • #27
dyn said:
That equations seems a bit strange to me. On the left hand side is a vector and the right hand side is a scalar.

I think my confusion is over what internal and external forces are. In the 2-body problem of a planet orbiting the sun i consider the gravitational forces between the 2 masses as internal forces. So with no external forces acting , this means no external torques , so i can automatically say that angular momentum is conserved in the 2-body problem. So i don't understand why books say that angular momentum is conserved due to the central nature of the force. As i understand it , the central nature of the force is irrelevant for the conserved angular momentum
It's of course nonsense :-((. The correct equation is
$$\mu \ddot{\vec{r}}=-G m_1 m_2 \frac{\vec{r}}{r^3}.$$
 
  • #28
jbriggs444 said:
If you like, it follows from an assumptions that pointlike object are pointlike

That just means that they have no spatial extension.

jbriggs444 said:
that they have no invisible arrows identifying a preferred orientation

That's an additional assumption. Where does ist come from?

jbriggs444 said:
that the force between two pointlike bodies is purely a function of the relative positions of the two bodies and their internal properties

That's another additional assumption. Where does it come from.

jbriggs444 said:
All of these assumptions are reasonable enough prior to the discovery of electromagnetism

Newton was at least aware of magnetic forces. The fact that they were not well understood that time was reason anough to be very careful with assumptions about forces in general.

jbriggs444 said:
If I am working in the framework of classical mechanics and electrostatics, I do assume it (1) and I expect pretty much everyone else to assume it (2) because it is fairly obvious (2a) as above and because the model works within its domain of applicability.

I still do not see how classical mechanics and electrostatics exclude forces that doesn't act in the direction from one point mass toward another unless you derive it from conservation of angular momentum (in the latter case it would be not an assumption but a conclusion).
 
  • #29
DrStupid said:
That's an additional assumption. Where does ist come from?
It's an assumption. Full stop. You cannot construct a theory entirely from provable assumptions.
 
  • #30
jbriggs444 said:
It's an assumption. Full stop.

That's not how physics works. Assumptions must be justified.
 
  • #31
DrStupid said:
That's not how physics works. Assumptions must be justified.
It is enough that their consequences are, in principle, falsifiable. That is how physics works.
 
  • #32
jbriggs444 said:
It is enough that their consequences are, in principle, falsifiable.

That also applies to the opposite assumptions. What makes you assumption better than assuming that there is no restriction for the possible direction of forces?
 
  • #33
DrStupid said:
That also applies to the opposite assumptions. What makes you assumption better than assuming that there is no restriction for the possible direction of forces?
The one is predictive and falsifiable. The other is neither.

It is usually better to have physical laws that predict something rather than physical laws that do not.

In Feynmans words: "First, we guess it (audience laughter), no, don’t laugh, that’s the truth. Then we compute the consequences of the guess, to see what, if this is right, if this law we guess is right, to see what it would imply and then we compare the computation results to nature or we say compare to experiment or experience, compare it directly with observations to see if it works."

If you can't falsify it, you can't do science with it.
 
  • #34
jbriggs444 said:
The one is predictive and falsifiable. The other is neither.

The other is at least as falsifiable as conservation of angular momentum. And of what avail is a predictable assumption if you have no idea if the prediction is correct or not? (And you have no idea if the assumption lacks proper justification.)

I still do not agree with your claim, that the 3rd law conserves angular momentum if you just assume the additional restriction of the direction of forces. It is your assumption that, together with the 3rd law, conserves angular momentum. Such a restriction must be justified (e.g. by experimental observations). If you just assume it than the resulting conservation of momentum is an assumption as well.
 
  • #35
The strong form of Newton III is needed to prove conservation of angular momentum via a torque argument, whilst you cannot prove it directly in this manner if only the weak form is used.

But conservation of angular momentum of an isolated system is a more fundamental concept, arising from rotational symmetry. It applies no matter whether the internal forces are central or not. I believe this is what @kuruman was referring to with the example about the dipoles.

I assume that any construction using arbitrary non-central forces in which the angular momentum of an isolated system is not conserved is unphysical. Please correct if this part is wrong!
 

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