Talk:Newton's theorem of revolving orbits

Confusing phrase
Does "twice as small" (in Precession of the Moon's orbit) mean half the size or twice the size? R.e.b. (talk) 04:57, 19 August 2008 (UTC)


 * Sorry, it means "half the size", roughly 1.5° predicted versus 3.0° observed. Please don't quote me on these numbers; I'll try to find the exact values and add them to the article.  Thank you very much for catching that! :) Willow (talk) 23:25, 19 August 2008 (UTC)

Illustrative example: Cotes' spirals
I think the formulae for k and lambda need checking.

For example, cosh of an angle starts with value 1 for angle zero, and increases indefinitely. cosh of an angle cannot go less than 1. This means that 1/r cannot go to zero, but can go indefinitely high. r, therefore, cannot go to infinity, but can only reach a certain maximum distance (although it can get ever smaller).Roo60 (talk) 00:30, 8 February 2009 (UTC)


 * This does not seem to contradict the article. For the moment I've removed your expert attention template, but if you can clarify what you think is wrong, I'll be happy to look into it. Ozob (talk) 00:17, 10 February 2009 (UTC)

I have gathered the various results stated in this section:

For this problem, there are 3 different solutions:
 * 1.) $$\frac{1}{r} = \frac{1}{b} \cos\ \left(\frac{\theta_2 - \theta_0}{k} \right)

$$, where the constant  $$k^2 = 1 - \frac{m \mu}{L_1^2}$$.


 * 2.) $$\frac{1}{r} = \frac{1}{b} \cosh\ \left(\frac{\theta_0 - \theta_2}{\lambda} \right)$$, where the constant $$\lambda^2 = -\frac{m \mu}{L_1^{2}} - 1$$.


 * 3.) $$\frac{1}{r} = A \theta_2 + \varepsilon$$, where A and ε are arbitrary constants.

For case 1, the possible values of the parameter k may range from zero to infinity, which corresponds to values of μ ranging from negative infinity up to the positive upper limit, L12/m.

For case 2, The possible values of λ range from zero to infinity, which corresponds to values of μ less than the negative number -L12/m.

Now, there seems to be an overlap of μ values for these two cases, between negative infinity and -L12/m. So, for the forces with μ values in this region, there are two possible orbits for the second particle. If I have learned my physics correctly, this can only happen in QM. Can someone please explain what's happening here? Thanks!--LaoChen (talk) 06:20, 10 August 2010 (UTC)


 * Yes, there's a typo. There was an extra minus sign in the expression for &lambda;. I've removed it, so the article is OK again. Incidentally, the two formulas are really instances of the same formula: Since cos ix = cosh x, you can deduce the second formula from the first (or vice versa) if you assume that the formula ought to remain true for all &mu;. Ozob (talk) 23:52, 11 August 2010 (UTC)

Problem with equation in Generalization section
I don't think the following force equation for the particle 2 is correct:

F_2(r_2) = \frac{a^3}{\left( 1 - b r_2 \right)^2} F_{1}\left( \frac{a r_2}{1 - b r_2} \right) + \frac{L^2}{mr_2^3} \left( 1 - k^2 \right) - \frac{bL^2}{mr_2^2} \,\!$$.

Take the case
 * $$a=0,\qquad k=1\,\!$$.

Then, particle 2's orbit is a circle with radius $$1/b\,\!$$ :
 * $$r_2(t)=1/b\,\!$$.

Also, the angular velocities for both particle 1 and particle are the same:
 * $$\theta_1(t)=\theta_2(t)/k=\theta_2(t)\,\!$$.

The angular momentum for particle 2 should be a constant for a central force:
 * $$L_2=mr_2^2\dot{\theta}_2\,\!$$.

So, the angular velocities for both particle 1 and particle are same constant
 * $$\dot{\theta}_1=\dot{\theta}_2= L_2 b^2/m\,\!$$.

Now, the force for particle 1 is arbitrary, let's say of the following inverse square distance form:
 * $$F_1(r_1)= - \mu/r_1^2\,\!$$.

So, the usual orbit for particle 1 should be an ellipse with radius $$r_1(t)\,\!$$ not been a constant. Then, the angular momentum for particle 1 is not constant either:
 * $$L_1=mr_1^2\dot{\theta}_1=L_2 b^2 r_1^2\,\!$$.

But, for central force, angular momentum should be a constant. There must be something wrong. Please help!--LaoChen (talk) 05:51, 9 September 2010 (UTC)

Can someone elaborate on the case of an imaginary coefficient?
Where it says ″By contrast, if k2 is less than one, F2−F1 is a positive number; the added inverse-cube force is repulsive″, I wonder what is the physical meaning of k2 being negative, which obviously implies that k is imaginary. Can some expert elaborate on this case, and explain how can k, the ratio between the angular speeds, be imaginary? Thanks a lot. — Preceding unsigned comment added by 94.209.165.16 (talk) 16:11, 22 September 2014 (UTC)
 * It's not "negative" but less than one. The value k (the article should probably not say it can be "any constant", but I'll look into that) is the ratio of the angular speed of one particle to another. As such it can be 0 (exactly the same speed) or any positive number (some multiple of the other particle's speed). Protonk (talk) 17:27, 22 September 2014 (UTC)

Irrelevant matter
I've tried, but can't at all see the relevance of all the stuff about ancient observations of planetary motion, retrogradation, mention of epicycles, to the subject of this article.

This article, in its essentials, is about a (very significant, to be sure) point within Newton's rational-mechanical analysis of orbits under different conceivable laws of central force – e.g. including exactly inverse-square, an inverse power law with an index a little more than 2, an index a little less than 2, and so on with other power laws. He shows how different power laws reveal themselves in the characteristics of the orbit, e.g. the rotational motion (or lack of it) of the direction of the long axis of a slightly elliptical orbit. One of his purposes was to provide a test, very sensitive for his time, by which observations could furnish a check on the reality and accuracy, or otherwise, of the inverse square law in the physical world.

The recent research explores numerous extensions of the applicable conditions in various ways; it shows among other things what happens if approximative assumptions made by Newton are not made; what happens if the restriction 'slightly' on the eccentricity of the ellipses (i.e. restriction to very small values of eccentricity) is removed.

It's not apparent how there is any real nexus here with the basics of planetary motion and retrogradation still less the epicycles. I'd suggest that for the basics and that part of the history, a wikilink to the relevant stuff would be clearer, and would also help underline how the topics are distinct. The present collection of historical stuff seems irrelevant and confusing in its aggregate in the context, rather than enlightening. Terry0051 (talk) 20:17, 1 October 2009 (UTC)
 * I concur. NOrbeck (talk) 08:33, 13 August 2010 (UTC)