Planetary orbits - the 2-body problem

[etotheipi]

If the forces between massive bodies don't satisfy lex 3, something else besides the massive bodies must be carrying angular momentum (as pointed out by @Ibix ). The but the angular momentum of an isolated system is conserved regardless.

Not necessarily! Yes, for electromagnetic interactions you can write the four momentum $P^i$ of the fields + the matter and conserve the associated angular momentum by setting the divergence of the integrand equal to zero.

But the Newtonian formalism does not prevent you from considering any other types of non-central forces for which the $\mathbf{L}_O = \sum_a \mathbf{r}_a \times \mathbf{p}_a$ is not an integral of the motion

 

[jbriggs444]

You can ONLY resolve this with formulae.

You can NEVER resolve matters of definition without using the definition(s).

 

[A.T.]

But the Newtonian formalism does not prevent you from considering any other types of non-central forces...

Maybe @dyn is not talking about anything that the Newtonian formalism doesn't explicitly prevent. Just saying that angular momentum is conserved in the real world.

 

[etotheipi]

Maybe @dyn is not talking about anything that the Newtonian formalism doesn't explicitly prevent. Just saying that angular momentum is conserved in the real world.

Perhaps, but the point is definitely worth stating! If you are trying to prove that this $\mathbf{L}$ thing is conserved for some system you're studying, at some point you're going to need to use that $(\mathbf{r}_a - \mathbf{r}_b) \times \mathbf{F}_{ab} = \mathbf{0}$ or similar. So it is important, as for any result, to understand what are the necessary assumptions. A priori there is no reason why this condition on the internal forces must hold!

 

[DrStupid]

What do you mean by "conclusion from conservation of momentum to conservation of angular momentum"?

Deriving conservation of angular momentum from conservation of momentum (or Newton's laws of motion).

@dyn's post was wrong and yours is too.

Are you really telling me that angular momentum is not conserved? Then please provide a proper reference.

 

[etotheipi]

Deriving conservation of angular momentum from conservation of momentum (or Newton's laws of motion).

In fact I never mentioned or wrote anything along those lines, so I don't know what you are referring to. But if you want then it's indeed possible to show that $\mathbf{L}$ is an integral using Newton's laws, yeah.

You can also just do it from symmetry principles, e.g. see post #51.

Are you really telling me that angular momentum is not conserved? Then please provide a proper reference.

Yes in the general case this $\mathbf{L} = \sum_a \mathbf{r}_a \times \mathbf{p}_a$ is not an integral of the motion, unless you also assume the internal forces are central.

How many times do I need to repeat the same thing until you understand?

 

[DrStupid]

In fact I never mentioned or wrote anything along those lines

That mean this is not from you:

If the two bodies are at positions $\mathbf{r}_a$ and $\mathbf{r}_b$ with respect to some origin $O$ and they exert forces $\mathbf{F}$ and $-\mathbf{F}$ respectively on each other, then $$\mathbf{G}_O = (\mathbf{r}_a - \mathbf{r}_b) \times \mathbf{F} = \frac{d\mathbf{L}_O}{dt}$$where $\mathbf{L}_O = m_a \mathbf{r}_a \times \mathbf{v}_a + m_b \mathbf{r}_b \times \mathbf{v}_b$. The quantity $\mathbf{L}_O$ is only a constant in the non-trivial case if $\mathbf{F} \parallel (\mathbf{r}_a - \mathbf{r}_b)$.

There must be something wrong with the forum software.

Yes

Your reference is missing as well.

 

[etotheipi]

Your reference is missing as well.

It's really not hard! This is ridiculous, it's literally a high school level argument\begin{align*}
\frac{d\mathbf{L}_O}{dt} &= \sum_a \frac{d}{dt} \left( \mathbf{r}_a \times \mathbf{p}_a \right) \\

&= \sum_a \left( \underbrace{\mathbf{v}_a \times \mathbf{p}_a}_{= \mathbf{0}} + \mathbf{r}_a \times \dot{\mathbf{p}}_a \right) \\ \\

&= \sum_a \mathbf{r}_a \times \mathbf{F}_a \\

&= \sum_b \sum_{a \neq b} \mathbf{r}_a \times \mathbf{F}_{ab} \\

&= \sum_b \sum_{a<b} (\mathbf{r}_a - \mathbf{r}_b) \times \mathbf{F}_{ab}
\end{align*}Now you may only generally conclude that $\dfrac{d\mathbf{L}_O}{dt} = \mathbf{0}$ if $\mathbf{F}_{ab} \parallel (\mathbf{r}_a - \mathbf{r}_b)$, i.e. if the internal forces are central.

 

[DrStupid]

It's really not hard!

It seems it is. Please refer to the Physics Forums Global Guidelines for acceptable references.

 

[berkeman]

It seems it is. Please refer to the Physics Forums Global Guidelines for acceptable references.

You need a reference for high school level math? Why? Do you have a reference that shows the reply is mistaken?

 

[dyn]

I just want to say thank you to everyone who replied in this thread. I appreciate your time and respect all your opinions and arguments

 

[fresh_42]

Extraordinary claims require extraordinary evidence!

... which is what the guidelines meant to communicate. However, there is an unspoken opposite of it:

Simple truths require pencil and paper to check.

E.g. I have no reference for, however, claim that ...

$$
\mathfrak{A(g)}=\{\alpha :\mathfrak{g}\longrightarrow \mathfrak{g}\,|\,\forall \,X,Y\in \mathfrak{g}\, : \,[\alpha (X),Y]+[X,\alpha (Y)]=0\} \text{ is a Lie algebra}
$$

... or at least none I would easily find. Nevertheless, all it needs to prove this statement are some applications of the Jacobi identity. Literally, everybody should be able to do this. And if someone wants proof, then I could either write down the few (boring) steps, as has been done in the above case, or simply say:
"A closer look at why $\mathfrak{Der(g)}$ is a Lie algebra shows, that the terms can be paired in such a way, that the required condition for $\mathfrak{A(g)}$ is a separate part of it."

You cannot demand references for some specific, nevertheless easy calculations.

 

[DrStupid]

You need a reference for high school level math?

No, I asked @etotheipi to provide a reference for his claim that angular momentum is not conserved. The high school level math he posted instead just shows that conservation of angular momentum does not follow from Newton's laws of motion without additional conditions. I already mentioned in #47 that this is something different and does not mean that angular momentum is not conserved. Thus, I'm still waiting for the reference.

 

[vanhees71]

This is really ridiculous. It is very clear, what a central interaction force is in the literature and for which forces angular momentum is conserved, namely precisely for central forces. I don't know, why the forum more and more has the tendency to discuss pseudo-problems and why we get more and more into a mode where you have to argue like a lawyer rather than a scientist. I have given the really simple proof, and @etotheipi has done so too. He is right in saying that this is high school level. If you need a reference take an arbitrary standard textbook on mechanics.

 

[fresh_42]

There is no need to discuss the distributive law or vector addition endlessly.

Thread closed.