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Update: Earth-Moon-Sun Trajectory equations
« on: April 25, 2019, 03:08:27 AM »
This update is in regards to a prior thread about the 3 body problem.

Summary
Prior discussion ended with a call for direct evidence of solutions to the 3 body problem, specifically to describe the motions of the Sun-Earth-Moon system under the central force model. Multiple references were provided, but a request was made for a synopsis which contained distilled results easily identified - rather than committing to several hundred pages of technical manuscript.

This Offering
I have procured a link to a concise summary that is intended for an upper division undergraduate physics student audience. The benefit of this approach is that it presents the nominal equations without additional theoretical applications often found in publications or research. Hence, it is my hope that this resource will prove useful for FEers who seek direct evidence that the 3 body trajectories are not only known, but reasonably simple and accessible to a scientific but not necessarily professional physicist audience.

As always, it is my pleasure to offer my time in fielding any questions you may have about this resource.

http://farside.ph.utexas.edu/teaching/celestial/Celestialhtml/node100.html
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Offline mitch

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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #1 on: April 25, 2019, 08:34:32 AM »
So this level of math eludes me.. Probably a stupid question but are you using these formulas to prove the earth is a sphere or flat? (Sorry I'm a noob)

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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #2 on: April 25, 2019, 09:22:22 AM »
So this level of math eludes me.. Probably a stupid question but are you using these formulas to prove the earth is a sphere or flat? (Sorry I'm a noob)

No apology necessary, and thanks for your question!

These equations describe the trajectories of the Earth, moon, and sun in our solar system. They are how, in part, eclipses can be predicted to such high accuracy (we know exactly when they will occur for the next several hundred years - and longer, if we wish to interpolate that far). Hope this helps  :)
The fact.that it's an old equation without good.demonstration of the underlying mechamism behind it makes.it more invalid, not more valid!

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Offline Tom Bishop

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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #3 on: April 25, 2019, 04:56:54 PM »
What you have posted are equations of motion, and how they may be written. "The equations of motion of the three bodies that make up the Earth-Moon-Sun system can be written..."

It is not a simulation of the three body problem. Please quote the section which shows that Newton's equations can actually result in a stable system with multiple bodies.

Your source admits that the three body problem has not been solved.

http://farside.ph.utexas.edu/teaching/celestial/Celestialhtml/node79.html

Quote
We saw earlier, in Section 2.9, that an isolated dynamical system consisting of two freely moving point masses exerting forces on one another--which is usually referred to as a two-body problem--can always be converted into an equivalent one-body problem. In particular, this implies that we can exactly solve a dynamical system containing two gravitationally interacting point masses, because the equivalent one-body problem is exactly soluble. (See Sections 2.9 and 4.16.) What about a system containing three gravitationally interacting point masses? Despite hundreds of years of research, no useful general solution of this famous problem--which is usually called the three-body problem--has ever been found.

The passage continues:

Quote
It is, however, possible to make some progress by severely restricting the problem's scope.

The next pages go on to talk about the restricted three body problem, which assumes that one of the bodies is massless. While still chaotic, the restricted three body problem doesn't fly apart. Those sections contain additional equations of motion. That is the current status of the three body problem: Odd scenarios in the attempt of making progress.

A description of equations of motion neither implies that the system will be stable, or that it will have the desired result.
« Last Edit: April 25, 2019, 05:10:11 PM by Tom Bishop »

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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #4 on: April 25, 2019, 05:17:57 PM »
So this level of math eludes me.. Probably a stupid question but are you using these formulas to prove the earth is a sphere or flat? (Sorry I'm a noob)

No apology necessary, and thanks for your question!

These equations describe the trajectories of the Earth, moon, and sun in our solar system. They are how, in part, eclipses can be predicted to such high accuracy (we know exactly when they will occur for the next several hundred years - and longer, if we wish to interpolate that far). Hope this helps  :)

You are in error. That is not how the eclipses are predicted.

https://wiki.tfes.org/Astronomical_Prediction_Based_on_Patterns

Quote
The Eclipses

In Chapter 11 of Earth Not a Globe its author gives us an overview of the eclipse calculations:

  “ Those who are unacquainted with the methods of calculating eclipses and other phenomena, are prone to look upon the correctness of such calculations as powerful arguments in favour of the doctrine of the earth's rotundity and the Newtonian philosophy, generally. One of the most pitiful manifestations of ignorance of the true nature of theoretical astronomy is the ardent inquiry so often made, "How is it possible for that system to be false, which enables its professors to calculate to a second of time both solar and lunar eclipses for hundreds of years to come?" The supposition that such calculations are an essential part of the Newtonian or any other theory is entirely gratuitous, and exceedingly fallacious and misleading. Whatever theory is adopted, or if all theories are discarded, the same calculations can be made. The tables of the moon's relative positions for any fraction of time are purely practical--the result of long-continued observations, and may or may not be connected with hypothesis. The necessary data being tabulated, may be mixed up with any, even the most opposite doctrines, or kept distinct from every theory or system, just as the operator may determine.

...The simplest method of ascertaining any future eclipse is to take the tables which have been formed during hundreds of years of careful observation; or each observer may form his own tables by collecting a number of old almanacks one for each of the last forty years: separate the times of the eclipses in each year, and arrange them in a tabular form. On looking over the various items he will soon discover parallel cases, or "cycles" of eclipses; that is, taking the eclipses in the first year of his table, and examining those of each succeeding year, he will notice peculiarities in each year's phenomena; but on arriving to the items of the nineteenth and twentieth years, he will perceive that some of the eclipses in the earlier part of the table will have been now repeated--that is to say, the times and characters will be alike. If the time which has elapsed between these two parallel or similar eclipses be carefully noted, and called a "cycle," it will then be a very simple and easy matter to predict any future similar eclipse, because, at the end of the "cycle," such similar eclipse will be certain to occur; or, at least, because such repetitions of similar phenomena have occurred in every cycle of between eighteen and nineteen years during the last several thousand years, it may be reasonably expected that if the natural world continues to have the same general structure and character, such repetitions may be predicted for all future time. The whole process is neither more nor less--except a little more complicated--than that because an express train had been observed for many years to pass a given point at a given second--say of every eighteenth day, so at a similar moment of every cycle or eighteenth day, for a hundred or more years to. come, the same might be predicted and expected. To tell the actual day and second, it is only necessary to ascertain on what day of the week the eighteenth or "cycle day" falls.

Tables of the places of the sun and moon, of eclipses, and of kindred phenomena, have existed for thousands of years, and were formed independently of each other, by the Chaldean, Babylonian, Egyptian, Hindoo, Chinese, and other ancient astronomers. Modern science has had nothing to do with these; farther than rendering them a little more exact, by averaging and reducing the fractional errors which a longer period of observation has detected. ” — Samuel Birley Rowbotham

Rowbotham provides pattern-based equations for finding the time, magnitude, and duration of the Lunar Eclipse at the end of Chapter 11.

The Royal Astronomer Sir Robert Ball, in his work The Story of the Heavens, on page 58, informs us:

  “ If we observe all the eclipses in a period of eighteen years, or nineteen years, then we can predict, with at least an approximation to the truth, all the future eclipses for many years. It is only necessary to recollect that in 6585 ⅓ days after one eclipse a nearly similar eclipse follows. For instance, a beautiful eclipse of the moon occurred on the 5th of December, 1881. If we count back 6585 days from that date, that is, 18 years and 2 days, we come to November 24th, 1863, and a similar eclipse of the moon took place then. …It was this rule which enabled the ancient astronomers to predict the occurrence of eclipses at a time when the motions of the moon were not understood nearly so well as we now know them. ”

Somerville in Physical Sciences pg. 46 states:

  “ No particular theory is required to calculate Eclipses, and the calculations may be made with equal accuracy, independent of every theory. ”

T.G. Ferguson in the Earth Review for September 1894, told us:

  “ No Doubt some will say, 'Well, how do the astronomers foretell the eclipses so accurately.' This is done by cycles. The Chinese for thousands of years have been able to predict the various solar and lunar eclipses, and do so now in spite of their disbelief in the theories of Newton and Copernicus. Keith says 'The cycle of the moon is said to have been discovered by Meton, an Athenian in B.C. 433,' then, of course, the globular theory was not dreamt of. ”

NASA Eclipse Website

Website URL: https://eclipse.gsfc.nasa.gov

If one visits NASA's eclipse website they will find that NASA explains eclipse prediction through the ancient Saros Cycle, rather than the Three Body Problem of astronomy.

From Resources -> Eclipses and the Saros (Archive) we read a description of the Saros Cycle:

  “ The periodicity and recurrence of eclipses is governed by the Saros cycle, a period of approximately 6,585.3 days (18 years 11 days 8 hours). It was known to the Chaldeans as a period when lunar eclipses seem to repeat themselves, but the cycle is applicable to solar eclipses as well. ”

The reader is encouraged to visit NASA's eclipse website and count how many times the Saros Cycle is mentioned, and then count how many times the Three Body Problem is mentioned.

Google Search Term: "saros" site:https://eclipse.gsfc.nasa.gov

No. of Results: 13,700

Google Search Term: "three body" site:https://eclipse.gsfc.nasa.gov

No. of Results: 2 (duplicate text)

  “ The distance of apogee does not vary by much month to month although the value of perigee can change quite a bit. Minimum vs. maximum apogee is a 0.6% spread and minimum vs. maximum perigee is a 3.9% spread. If Newton couldn't solve the three-body problem I certainly can't ”

The Three Body Problem refers to the greatest problem in the history of astronomy. It is the inability of science to simulate or recreate a model of the Sun-Earth-Moon system. It is for this reason that pattern-based methods must be used for prediction in astronomy.

Read any astronomy book and the same will be seen:

A TEXT-BOOK OF ASTRONOMY
by George C. Comstock
Director of the Washburn Observatory and
Professor of Astronomy in the
University of Wisconson

http://www.gutenberg.org/files/34834/34834-h/34834-h.htm


(click to enlarge)

One should notice that there is nothing about the three body problem, the geometry of the sun-earth-moon system, or Newton's equations as being the basis for the eclipse predictions.
« Last Edit: April 25, 2019, 10:36:47 PM by Tom Bishop »

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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #5 on: April 25, 2019, 06:13:12 PM »
So ... why do you think the author states "no useful general solution" has been found, as opposed to simply stating "no solution" has been found?

Why do you think he included those two keywords in particular?
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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #6 on: April 25, 2019, 06:15:20 PM »
So that would possibly mean that there is a) a useless general solution or b) a useful specific solution.
BobLawBlah.

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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #7 on: April 25, 2019, 06:19:34 PM »
So ... why do you think the author states "no useful general solution" has been found, as opposed to simply stating "no solution" has been found?

Why do you think he included those two keywords in particular?

The only solutions involve very specific, odd scenarios.

https://www.askamathematician.com/2011/10/q-what-is-the-three-body-problem/

Quote
Q: What is the three body problem?

The three body problem is to exactly solve for the motions of three (or more) bodies interacting through an inverse square force (which includes gravitational and electrical attraction).

The problem with the 3-body problem is that it can’t be done, except in a very small set of frankly goofy scenarios (like identical planets following identical orbits).

From a New Scientist article titled "Infamous three-body problem has over a thousand new solutions":

https://www.newscientist.com/article/2148074-infamous-three-body-problem-has-over-a-thousand-new-solutions/

Quote
The new solutions were found when researchers at Shanghai Jiaotong University in China tested 16 million different orbits using a supercomputer.

...

Perhaps the most important application of the three-body problem is in astronomy, for helping researchers figure out how three stars, a star with a planet that has a moon, or any other set of three celestial objects can maintain a stable orbit.

But these new orbits rely on conditions that are somewhere between unlikely and impossible for a real system to satisfy. In all of them, for example, two of the three bodies have exactly the same mass and they all remain in the same plane.


Knot-like paths

In addition, the researchers did not test the orbits’ stability. It’s possible that the tiniest disturbance in space or rounding error in the equations could rip the objects away from one another.

“These orbits have nothing to do with astronomy, but you’re solving these equations and you’re getting something beautiful,” says Vanderbei.

...

Aside from giving us a thousand pretty pictures of knot-like orbital paths, the new three-body solutions also mark a starting point for finding even more possible orbits, and eventually figuring out the whole range of winding paths that three objects can follow around one another.

...

“This is kind of the zeroth step. Then the question becomes, how is the space of all possible positions and velocities filled up by solutions?” says Richard Montgomery at the University of California, Santa Cruz. “These simple orbits are kind of like a skeleton to build the whole system up from.”

As suggested, the field of Celestial Mechanics is still on step zero—the stone age. The found orbits are nothing like heliocentric astronomy and there will be an attempt to use them as a skeleton to "build the whole system up from."
« Last Edit: April 25, 2019, 06:46:34 PM by Tom Bishop »

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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #8 on: April 25, 2019, 07:23:14 PM »
What you have posted are equations of motion, and how they may be written. "The equations of motion of the three bodies that make up the Earth-Moon-Sun system can be written..."

It is not a simulation of the three body problem. Please quote the section which shows that Newton's equations can actually result in a stable system with multiple bodies.

Your source admits that the three body problem has not been solved.

http://farside.ph.utexas.edu/teaching/celestial/Celestialhtml/node79.html

Quote
We saw earlier, in Section 2.9, that an isolated dynamical system consisting of two freely moving point masses exerting forces on one another--which is usually referred to as a two-body problem--can always be converted into an equivalent one-body problem. In particular, this implies that we can exactly solve a dynamical system containing two gravitationally interacting point masses, because the equivalent one-body problem is exactly soluble. (See Sections 2.9 and 4.16.) What about a system containing three gravitationally interacting point masses? Despite hundreds of years of research, no useful general solution of this famous problem--which is usually called the three-body problem--has ever been found.

The passage continues:

Quote
It is, however, possible to make some progress by severely restricting the problem's scope.

The next pages go on to talk about the restricted three body problem, which assumes that one of the bodies is massless. While still chaotic, the restricted three body problem doesn't fly apart. Those sections contain additional equations of motion. That is the current status of the three body problem: Odd scenarios in the attempt of making progress.

A description of equations of motion neither implies that the system will be stable, or that it will have the desired result.

I am sorry, but your reply is demonstrably false. There is no mathematical statement which makes one object massless, and it is evident in the equations this is not true. What you are referencing is the condition that the sun’s mass is much greater than the moon’s, which has a mathematical implication in the solutions - which is why the “massless limit” is discussed. But obviously physicists do not compute orbits of massless planets - that is absurd.

Severely restricting the problem’s scope is to apply the general theory to the earth moon sun system in particular, and to relax the corrections which would explain stable perturbations from tidal forces - which are simply beyond the scope of an undergraduate audience. This is why the word “scope” is even used. For details on those items, please see my previous reference.

The varied exotic orbits that you have found demonstrate, as I have said, how sophisticated our understanding is in these matters. Not only can we model the relatively simple solar system, but incredibly chaotic and intricate orbits as well. It is plainly bizarre that you interpret our ability to model such complex orbits as evidence of a celestial mechanics infancy, especially when describing the solar system orbits is so trivial in comparison and has been done for much longer.

Also, you requested the exact equations, and I have provided them.

Twice.

If you now wish to move the goalposts and insist on a simulation, well you can do that on your own time. I do not work for free, Tom, and if you wish me to perform physics computation for you, then you will pay me for the privilege enjoyed. And I probably incur an honorarium that eclipses your budget.

Fortunately, plotting the equations or simulating their evolutions is probably a task manageable by the FES. I wish you luck in that endeavor.

Until then, this matter has been resolved.

Oh P.S., I didn’t say the equations were necessarily used by everyone to predict the eclipses - heck, ancient cultures could do these predictions using pattern recognition strategies. What I actually said was that the equations DO predict the eclipses (obviously, because they describe the trajectories). Other methods do exist which some folks use instead :)
« Last Edit: April 25, 2019, 07:42:50 PM by QED »
The fact.that it's an old equation without good.demonstration of the underlying mechamism behind it makes.it more invalid, not more valid!

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Offline Tom Bishop

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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #9 on: April 25, 2019, 10:00:23 PM »
Quote from: QED
I am sorry, but your reply is demonstrably false. There is no mathematical statement which makes one object massless, and it is evident in the equations this is not true. What you are referencing is the condition that the sun’s mass is much greater than the moon’s, which has a mathematical implication in the solutions - which is why the “massless limit” is discussed. But obviously physicists do not compute orbits of massless planets - that is absurd.

Severely restricting the problem’s scope is to apply the general theory to the earth moon sun system in particular, and to relax the corrections which would explain stable perturbations from tidal forces - which are simply beyond the scope of an undergraduate audience. This is why the word “scope” is even used. For details on those items, please see my previous reference.

Look into George Hill's work on the Three Body Problem and heliocentric orbits. The only way Hill was able to make any progress at all was by using the Restricted Three Body Problem, where one of the bodies was of zero or negligable mass. Even then, the body was still chaotic. The only benefit of the Restricted Three Body Problem and the Mass-less moon is that the moon is no longer ejected from the system, as it usually is. It is confined to what is known as "Hill's Region".

http://www.scholarpedia.org/article/Three_body_problem



We can see the crazy chaotic moon. It even makes a u-turn in mid orbit.

From the text that accompanies the image:

Quote
The simplest case:

It occurs when, the Jacobi constant being negative and big enough, the zero mass body (we shall still call it the Moon) moves in a component of the Hill region which is a disc around one of the massive bodies (the Earth). This fact already implies Hill's rigorous stability result: for all times such a Moon would not be able to escape from this disc. Nevertheless this does not prevent collisions with the Earth.

"Zero mass body" -- One of the bodies in the restricted three body problem is of zero mass.

"Nevertheless this does not prevent collisions with the earth" -- It's still chaotic, even in that simplified version.

Newtonian mechanics most certainly does not naturally default to the heliocentric system of Copernicus.

Quote from: QED
I do not work for free, Tom, and if you wish me to perform physics computation for you, then you will pay me for the privilege enjoyed. And I probably incur an honorarium that eclipses your budget.

That is not necessary. The Three Body Problem has already been studied at depth by many mathematicians over several hundred years. History's Greatest Mathematicians > You. The conclusion is that there are no good solutions. One only needs to read the material on the subject to see that.
« Last Edit: April 25, 2019, 10:33:14 PM by Tom Bishop »

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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #10 on: April 26, 2019, 03:44:03 AM »
Quote from: QED
I am sorry, but your reply is demonstrably false. There is no mathematical statement which makes one object massless, and it is evident in the equations this is not true. What you are referencing is the condition that the sun’s mass is much greater than the moon’s, which has a mathematical implication in the solutions - which is why the “massless limit” is discussed. But obviously physicists do not compute orbits of massless planets - that is absurd.

Severely restricting the problem’s scope is to apply the general theory to the earth moon sun system in particular, and to relax the corrections which would explain stable perturbations from tidal forces - which are simply beyond the scope of an undergraduate audience. This is why the word “scope” is even used. For details on those items, please see my previous reference.

Look into George Hill's work on the Three Body Problem and heliocentric orbits. The only way Hill was able to make any progress at all was by using the Restricted Three Body Problem, where one of the bodies was of zero or negligable mass. Even then, the body was still chaotic. The only benefit of the Restricted Three Body Problem and the Mass-less moon is that the moon is no longer ejected from the system, as it usually is. It is confined to what is known as "Hill's Region".

http://www.scholarpedia.org/article/Three_body_problem



We can see the crazy chaotic moon. It even makes a u-turn in mid orbit.

From the text that accompanies the image:

Quote
The simplest case:

It occurs when, the Jacobi constant being negative and big enough, the zero mass body (we shall still call it the Moon) moves in a component of the Hill region which is a disc around one of the massive bodies (the Earth). This fact already implies Hill's rigorous stability result: for all times such a Moon would not be able to escape from this disc. Nevertheless this does not prevent collisions with the Earth.

"Zero mass body" -- One of the bodies in the restricted three body problem is of zero mass.

"Nevertheless this does not prevent collisions with the earth" -- It's still chaotic, even in that simplified version.

Newtonian mechanics most certainly does not naturally default to the heliocentric system of Copernicus.

Quote from: QED
I do not work for free, Tom, and if you wish me to perform physics computation for you, then you will pay me for the privilege enjoyed. And I probably incur an honorarium that eclipses your budget.

That is not necessary. The Three Body Problem has already been studied at depth by many mathematicians over several hundred years. History's Greatest Mathematicians > You. The conclusion is that there are no good solutions. One only needs to read the material on the subject to see that.

I have provided several references that detail solutions which demonstrate your above claim is false. You clearly did not read them, or do not want to acknowledge them. If baffles me why.

While a general solution to the 3 body problem doesn’t exist, this doesn’t matter. A general solution would contain every single orbit possible for a three body system. That’s what “general” means.

Example: a general quadratic is ax^2+bx+c. Every single quadratic is represented here.

A particular quadratic: x^2.

You are claiming that, because we do not have the general quadratic for the 3 body problem, we do not have the particular quadratic for the EMS system.

This is simply wrong, I have posted the equations to prove it, and you just do not understand mathematics enough to delineate general from specific solutions to differential equations. I have offered to help you with this. That offer still stands. 

The EMS system is a particular solution, you see. And although a general solution has not been found, dozens of particular ones have been found: the EMS, orbits that are simpler, orbits more complicated, even exotic orbits - some of which you have even referenced. You are not assembling all this information together correctly, that’s all! You’re just misunderstanding what it is telling you. You’re not stupid, or anything else, you just don’t have enough math background to interpret this all alone. Let me help you.

Claiming that the particular equations do not exist because a general solution cannot be found is simply a misunderstanding of what a general solution means and of basic mathematics in general.

I am losing confidence that you will ever learn this simple lesson, probably because you refuse to - but certainly not because you couldn’t. The choice is yours.

I shall continue to post the exact equations for the EMS orbits every time you claim that they somehow do not exist, and re-explain the above for who ever is reading it.

The audience of FEers and REers in attendance can judge for themselves.

Bye for now.
« Last Edit: April 26, 2019, 04:01:42 AM by QED »
The fact.that it's an old equation without good.demonstration of the underlying mechamism behind it makes.it more invalid, not more valid!

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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #11 on: April 26, 2019, 02:22:20 PM »
QED, the Sun-Earth-Moon system isn't a spacial case that works. You have provided zero sources or citations which contradict the sources which say that the only ones which work are the odd scenarios.

From https://www.askamathematician.com/2011/10/q-what-is-the-three-body-problem/ --

Quote
Q: What is the three body problem?

The three body problem is to exactly solve for the motions of three (or more) bodies interacting through an inverse square force (which includes gravitational and electrical attraction).

The problem with the 3-body problem is that it can’t be done, except in a very small set of frankly goofy scenarios (like identical planets following identical orbits).

We see the same conclusions everywhere we look. From National Cheng-Kung University we find a slideshow from a lecture: The Three Body Problem from a variational point of view

Start of Slideshow:



Towards the end:

"Recent Progress"



That was the recent progress for the matter at the time of the slideshow's creation.


From a book titled "Three Body Dynamics and Its Applications to Exoplanets" by Zdzislaw Musielak and Billy Quarles we find the following on p.82

Quote
5.3.2 The Sun-Earth-Moon System

The three-body problem in terms of the Sun-Earth-Moon system was a singular case that posed a challenge for Newton's universal law of gravity. He had developed algorithms to solve Kepler's equation and thereby the two-body problem with great success. The approach taken by Newton was that the Sun-Earth potential should dominate and treated the Moon as a test particle, which was not successful in determining stable solutions (~8% uncertainties). The problem was that the mass of the Moon is non-negligible (0.0167 M) relative to the mass of the Earth. Moreover the closeness of the Moon to the Earth is enough to induce reaction forces that are not accounted for in the restricted problem.

However, others have made progress in determining stability conditions for this particular problem, which is often referred to as Hill's problem (Hill 1877, 1878); see also Chaps. 2 and 3. Hill's problem is applicable to three-body problems where...

Hill's problem that saw earlier is portrayed as the current status of the Sun-Earth-Moon system.

From Summary and Concluding Remarks:

Quote
The three-body problem is one of the best known scientific problems, and there are three main reasons. First, the problem has fundamental importance in science, mathematics and engineering. Second, the problem is truly difficult to solve, which has been manifested by thousands of papers written by numerous outstanding researchers, who have tried (very) hard but failed. Third, the problem can be stated simple enough, so non-experts can understand it and appreciate its importance. As described in this book, all attempts to solve this problem have enriched celestial and classical mechanics, and mathematics with many new methods, powerful theorems, and novel ideas that are already being used in applications in different fields of research.

The material presented in this book covers in detail the general, circular restricted and elliptical restricted three-body problem as well as Hill's problems... The presented cases and discussed results intend to show the richness of problems that can be investigated by using the available numerical codes and modern powerful computers.

Once again the same is stated. Doesn't work. Nowhere is it stated that "actually, it works if you do it this way..." That is a complete imagination, without source, and not backed by the people studying the subject.
« Last Edit: April 26, 2019, 03:01:46 PM by Tom Bishop »

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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #12 on: April 26, 2019, 02:33:26 PM »
QED, the Sun-Earth-Moon system isn't a spacial case that works. You have provided zero sources or citations which directly state that the only ones which work are the odd scenarios.

https://www.askamathematician.com/2011/10/q-what-is-the-three-body-problem/ (Archive)

  “ Q: What is the three body problem?

The three body problem is to exactly solve for the motions of three (or more) bodies interacting through an inverse square force (which includes gravitational and electrical attraction).

The problem with the 3-body problem is that it can’t be done, except in a very small set of frankly goofy scenarios (like identical planets following identical orbits).

We see the same conclusions everywhere we look. From National Cheng-Kung University we find a slideshow from a lecture: The Three Body Problem from a variational point of view

Start of Slideshow:



Towards the end:

"Recent Progress"




From a book titled "Three Body Dynamics and Its Applications to Exoplanets" by Zdzislaw Musielak, Billy Quarles we find the following on p.82

Quote
5.3.2 The Sun-Earth-Moon System

The three-body problem in terms of the Sun-Earth-Moon system was a singular case that posed a challenge for Newton's universal law of gravity. He had developed algorithms to solve Kepler's equation and thereby the two-body problem with great success. The approach taken by Newton was that the Sun-Earth potential should dominate and treated the Moon as a test particle, which was not successful in determining stable solutions (~8% uncertainties). The problem was that the mass of the Moon is non-negligible (0.0167 Mg) relative to the mass of the Earth. Moreover the closeness of the Moon to the Earth is enough to induce reaction forces that are not accounted for in the restricted problem.

However, others have made progress in determining stability conditions for this particular problem, which is often referred to as Hill's problem (Hill 1877, 1878); see also Chaps. 2 and 3. Hill's problem is applicable to three-body problems where...

Hill's problem that saw earlier is portrayed as the current status of the Sun-Earth Moon system.

From the Summary and Concluding Remarks:

From Summary and Concluding Remarks

Quote
The three-body problem is one of the best known scientific problems, and there are three main reasons. First, the problem has fundamental importance in science, mathematics and engineering. Second, the problem is truly difficult to solve, which has been manifested by thousands of papers written by numerous outstanding researchers, who have tried (very) hard but failed. Third, the problem can be stated simple enough, so non-experts can understand it and appreciate its importance. As described in this book, all attempts to solve this problem have enriched celestial and classical mechanics, and mathematics with many new methods, powerful theorems, and novel ideas that are already being used in applications in different fields of research.

The material presented in this book covers in detail the general, circular restricted and elliptical restricted three-body problem as well as Hill's problems... The presented cases and discussed results intend to show the richness of problems that can be investigated by using the available numerical codes and modern powerful computers.

Once again the same is stated. Doesn't work. Nowhere is it stated that "actually, it works if you do it this way..." That is a complete imagination, without source.

Well sure it is! The EMS have particular masses, angular momentum, and energies. The three body problem for them has been solved, is rudimentary, and the equations which state this are found at the beginning of this thread.

I understand your confusion. You keep finding sources where mathematicians or physicists are saying that general solutions to the three body problem can’t be found. So you go: see the evidence is right here! You’re trying to trick me QED!

But it’s no trick. You are effectively debating semantics with yourself and don’t see it. The EMS system is a particular 3 body scenario, and you are simply confusing sources that discuss the general solutions and believing that they are taking about particular ones.

I suppose confirmation bias plays a role. I also suppose if I had spent significant time on an effort, only to learn it was a non-sequitur, I would be rather stubborn against accepting it too.

No matter how many internet sources you misinterpret, it will still seem paltry when compared to the fact that every day, across the world, undergraduates are being taught and using the very equations that you deny exist.

The best thing you could do is take a physics class and learn it, so that you become a more informed debater. Let me know how I can help!
The fact.that it's an old equation without good.demonstration of the underlying mechamism behind it makes.it more invalid, not more valid!

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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #13 on: April 26, 2019, 04:32:20 PM »
In The Physics Problem that Isaac Newton Couldn't Solve its author Robert Scherrer, Chair of the Department of Physics and Astronomy at Vanderbilt University, says otherwise:

Quote
There's a physics problem so difficult, so intractable, that even Isaac Newton, undoubtedly the greatest physicist who ever lived, couldn't solve it. And it's defied everyone else's attempts ever since then.

This is the famous three-body problem. When Newton invented his theory of gravity, he immediately set to work applying it to the motions of the planets in the solar system. If you have a planet orbiting a much larger body, like the sun, and the orbit is circular, then the problem is easy to solve -- it's something that's done in a high school physics class.

But a circular orbit isn't the most general possibility, and sometimes one body isn't much smaller than the object it orbits (think of the Moon going around the Earth). This more complicated case can still be solved -- Newton showed that the two bodies orbit their common center of mass in elliptical orbits. In fact, this prediction of elliptical orbits really cemented the case for Newton's theory of gravity. The calculation is a lot trickier than for circular orbits, but we still throw it at undergraduate physics majors in their second or third year.

Now add a third body, and everything falls apart. The problem goes from one that a smart undergraduate can tackle to one that has defied solution for 400 years.

Caltech physicist Sean Carrol says the same on his page N-Bodies. A history is provided:

Quote
Henri Poincare was a favorite to win the prize, and he submitted an essay that demonstrated the stability of planetary motions in the three-body problem (actually the “restricted” problem, in which one test body moves in the gravitational field generated by two others). In other words, without knowing the exact solutions, we could at least be confident that the orbits wouldn’t go crazy; more technically, solutions starting with very similar initial conditions would give very similar orbits. Poincare’s work was hailed as brilliant, and he was awarded the prize.

But as his essay was being prepared for publication in Acta Mathematica, a couple of tiny problems were pointed out by Edvard Phragmen, a Swedish mathematician who was an assistant editor at the journal. Gosta Mittag-Leffler, chief editor, forwarded Phragmen’s questions to Poincare, asking him to fix up these nagging issues before the prize essay appeared in print. Poincare went to work, but discovered to his consternation that one of the tiny problems was in fact a profoundly devastating possibility that he hadn’t really taken seriously. What he ended up proving was the opposite of his original claim — three-body orbits were not stable at all.  3-body orbits Not only were orbits not periodic, they didn’t even approach some sort of asymptotic fixed points.

...

However, just because the general solution to the three-body (and more-body) problem is chaotic, doesn’t mean we can’t find special exact solutions in highly-symmetric conditions, and that’s just what Cris Moore and Michael Nauenberg have recently been doing.

It doesn't work, and there are only some highly-symmetric special cases.

Nowhere is it mentioned that "actually it can be done if we do it this way..." or "mathematicians just do it the easy way by..." That most certainly would be brought up in these texts. No one is telling this story. I can only encourage you to back up your claims with appropriate sources.
« Last Edit: April 26, 2019, 04:44:30 PM by Tom Bishop »

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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #14 on: April 26, 2019, 11:38:18 PM »
In The Physics Problem that Isaac Newton Couldn't Solve its author Robert Scherrer, Chair of the Department of Physics and Astronomy at Vanderbilt University, says otherwise:

Quote
There's a physics problem so difficult, so intractable, that even Isaac Newton, undoubtedly the greatest physicist who ever lived, couldn't solve it. And it's defied everyone else's attempts ever since then.

This is the famous three-body problem. When Newton invented his theory of gravity, he immediately set to work applying it to the motions of the planets in the solar system. If you have a planet orbiting a much larger body, like the sun, and the orbit is circular, then the problem is easy to solve -- it's something that's done in a high school physics class.

But a circular orbit isn't the most general possibility, and sometimes one body isn't much smaller than the object it orbits (think of the Moon going around the Earth). This more complicated case can still be solved -- Newton showed that the two bodies orbit their common center of mass in elliptical orbits. In fact, this prediction of elliptical orbits really cemented the case for Newton's theory of gravity. The calculation is a lot trickier than for circular orbits, but we still throw it at undergraduate physics majors in their second or third year.

Now add a third body, and everything falls apart. The problem goes from one that a smart undergraduate can tackle to one that has defied solution for 400 years.

Caltech physicist Sean Carrol says the same on his page N-Bodies. A history is provided:

Quote
Henri Poincare was a favorite to win the prize, and he submitted an essay that demonstrated the stability of planetary motions in the three-body problem (actually the “restricted” problem, in which one test body moves in the gravitational field generated by two others). In other words, without knowing the exact solutions, we could at least be confident that the orbits wouldn’t go crazy; more technically, solutions starting with very similar initial conditions would give very similar orbits. Poincare’s work was hailed as brilliant, and he was awarded the prize.

But as his essay was being prepared for publication in Acta Mathematica, a couple of tiny problems were pointed out by Edvard Phragmen, a Swedish mathematician who was an assistant editor at the journal. Gosta Mittag-Leffler, chief editor, forwarded Phragmen’s questions to Poincare, asking him to fix up these nagging issues before the prize essay appeared in print. Poincare went to work, but discovered to his consternation that one of the tiny problems was in fact a profoundly devastating possibility that he hadn’t really taken seriously. What he ended up proving was the opposite of his original claim — three-body orbits were not stable at all.  3-body orbits Not only were orbits not periodic, they didn’t even approach some sort of asymptotic fixed points.

...

However, just because the general solution to the three-body (and more-body) problem is chaotic, doesn’t mean we can’t find special exact solutions in highly-symmetric conditions, and that’s just what Cris Moore and Michael Nauenberg have recently been doing.

It doesn't work, and there are only some highly-symmetric special cases.

Nowhere is it mentioned that "actually it can be done if we do it this way..." or "mathematicians just do it the easy way by..." That most certainly would be brought up in these texts. No one is telling this story. I can only encourage you to back up your claims with appropriate sources.

Again, these references are discussing the general solution.

The general solution to the two body problem is easy. For the three body it has not been found.

Again, the EMS system is not a general solution, and the particular orbits have been found. See the top of this thread for direct evidence.

You continue to (a) confuse the general solution for differential equations for specific solutions, and (b) misinterpret the information you are trying to assemble.

My only advice at this point is to take a class on differential equations. An approach of just misinterpreting it HARDER probably won’t help you.

Plus, in this last reply you are treating an absence of evidence as evidence for the opposite. You will be able to find miles of text that does not “tell this story,” just like you can find miles of text that doesn’t tell any other story of your choosing.

I searched for fifteen minutes and collected nine sources that “tell my story.” If you cannot find any for your own then it is probably just a selection bias.

I have provided 2 sources for your preview - in this thread.  The equations in them describe the specific trajectories of the EMS system. They’re still there, and have not gone away.

If you’d like me to take you through those equations I might be persuaded to invest that time. Especially for the benefit of others who might read it.

I am sorry that you cannot come to terms with this. If I can think of additional ways to help you then I will not hesitate.
« Last Edit: April 26, 2019, 11:46:27 PM by QED »
The fact.that it's an old equation without good.demonstration of the underlying mechamism behind it makes.it more invalid, not more valid!

- Tom Bishop

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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #15 on: April 27, 2019, 02:54:51 PM »
Ask a Mathematician:

"The problem with the 3-body problem is that it can’t be done, except in a very small set of frankly goofy scenarios (like identical planets following identical orbits).”

Sean Carrol:

"However, just because the general solution to the three-body (and more-body) problem is chaotic, doesn’t mean we can’t find special exact solutions in highly-symmetric conditions, and that’s just what Cris Moore and Michael Nauenberg have recently been doing."

Wikipedia listing of special-case solutions: https://en.m.wikipedia.org/wiki/Three-body_problem#Special-case_solutions

It's not there. There is no description of what you are claiming from any source. You appear to have made something up. You have not provided evidence for your clam that the Sun-Earth-Moon system is a special case.
« Last Edit: April 27, 2019, 03:06:24 PM by Tom Bishop »


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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #17 on: April 27, 2019, 03:51:15 PM »
Ask a Mathematician:

"The problem with the 3-body problem is that it can’t be done, except in a very small set of frankly goofy scenarios (like identical planets following identical orbits).”

Sean Carrol:

"However, just because the general solution to the three-body (and more-body) problem is chaotic, doesn’t mean we can’t find special exact solutions in highly-symmetric conditions, and that’s just what Cris Moore and Michael Nauenberg have recently been doing."

Wikipedia listing of special-case solutions: https://en.m.wikipedia.org/wiki/Three-body_problem#Special-case_solutions

It's not there. There is no description of what you are claiming from any source. You appear to have made something up. You have not provided evidence for your clam that the Sun-Earth-Moon system is a special case.

Your quotes are precisely correct. They are not mistaken, and the reason why you think that this somehow prohibits EMS trajectories from being found is because you do not understand the information you are assembling. Read the sources at the top of this thread. They provide exactly what you are looking for. They are evidence.
The fact.that it's an old equation without good.demonstration of the underlying mechamism behind it makes.it more invalid, not more valid!

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Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #18 on: April 27, 2019, 04:00:54 PM »
My only advice at this point is to take a class on differential equations.

https://forum.tfes.org/index.php?topic=10175.msg160183#msg160183

https://forum.tfes.org/index.php?topic=4310.msg84969#msg84969

https://forum.tfes.org/index.php?topic=4310.msg84958#msg84958

In regards to your references.

KAM theory consequence: the motions are quasi-periodic. This is absolutely fine and fits well with how we understand the solar system. Indeed, the solar system is only stable in time frames that humans care about. With sufficient time, the orbits will degrade even if the Sun was eternal. In fact, we can compute how many orbits it will take!

Your second reference is also for KAM, and does not add any value to your first.

Your third reference is an opinion piece, references KAM yet again, and contains claims that are questionable.

Thank you for providing additional support for RE astronomy in the form of referencing a theorem that we use to information outnumber predictions!
The fact.that it's an old equation without good.demonstration of the underlying mechamism behind it makes.it more invalid, not more valid!

- Tom Bishop

We try to represent FET in a model-agnostic way

- Pete Svarrior

Re: Update: Earth-Moon-Sun Trajectory equations
« Reply #19 on: April 27, 2019, 05:41:53 PM »
KAM theory consequence: the motions are quasi-periodic. This is absolutely fine and fits well with how we understand the solar system.

But it can't be fine.

All Hamiltonian systems which are not integrable are chaotic.

Since the solar system is not integrable, and experiences unpredictable small perturbations, it cannot lie permanently on a KAM torus, and is thus chaotic.

KAM theory is valid for "sufficiently" small perturbations.

In reality, the perturbations in the solar system are far too large to apply KAM theory.

So, the mathematicians have to rely on computing Lyapunov exponents, in order to try to predict any region of instability/chaos.

Jack Wisdom (MIT): It is not possible to exclude the possibility that the orbit of the Earth will suddenly exhibit similar wild excursions in eccentricity.

Even measuring initial conditions of the system to an arbitrarily high, but finite accuracy, we will not be able to describe the system dynamics "at any time in the past or future". To predict the future of a chaotic system for arbitrarily long times, one would need to know the initial conditions with infinite accuracy, and this is by no means possible.

Lyapunov exponents and symplectic integration.

Let d(t) be the distance between two solutions, with d(0) being their initial separation. Then d(t) increases approximately as d(0)eλt in a chaotic system, where λ is the Lyapunov exponent. The inverse of the Lyapunov exponent, 1/λ, is called the Lyapunov time, and measures how long it takes two nearby solutions to diverge by a factor of e.

Sussman and Wisdom's 1992 integration of the entire solar system displayed a disturbing dependence on the timestep of the integration (measurement of the Lyapunov time).

Thus, different researchers who draw their initial conditions from the same ephemeris at different times can find vastly different Lyapunov timescales.

Wayne Hayes, UC Irvine

To show the importance and the dependence on the sensitivity of the initial conditions of the set of differential equations, an error as small as 15 meters in measuring the position of the Earth today would make it impossible to predict where the Earth would be in its orbit in just over 100 million years' time.

“The word ‘chaotic’ summarizes many fundamental concepts characterizing
a dynamical system such as complex predictability and stability. But above
all, it acts as a warming of the difficulties which are likely to arise when trying to
obtain a reliable picture of its past and future evolution. As an example, a
commonly accepted definition states that a system is ‘unstable’ if the trajectories of
two points that initially are arbitrarily close . . . diverge quickly in time. This has
strong implications, as small uncertainties in initial conditions . . . might [also] be
consistent with completely different future trajectories: The conclusion is that we
can exactly reproduce the motion of a chaotic system only if WE KNOW, WITH
ABSOLUTE PRECISION, THE INITIAL CONDITIONS – A STATEMENT
THAT, IN PRACTICE, CAN NEVER BE TRUE."

Alessandra Celletti, Ettore Perozzi, Celestial Mechanics: The Waltz of the Planets

Let us take a closer look the chaotic dynamics of planetary formation; thus, a clear indication that the initial conditions cannot be predicted with accuracy (as we have seen, a mere 15 meters difference in the data will have catastrophic consequences upon the calculations).

OFFICIAL SCIENCE INFORMATION

Four stages of planetary formation

Initial stage: condensation and growth of grains in the hot nebular disk

Early stage: growth of grains to kilometer-sized planetesimals

Middle stage: agglomeration of planetesimals

Late stage: protoplanets


For the crucial stages, the initial and early stages, prediction becomes practically impossible.

As if this wasn't enough, we have absolute proof that in the age of modern man planet Earth underwent sudden pole shifts (heliocentrical version), thus making null and void any integration of the solar system/Lyapunov exponents calculations which do not take into account such variations of the system's parameters:

http://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1635693#msg1635693

http://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1546053#msg1546053

Let me show you what sensitive dependence on initial conditions means, using one of the most famous examples: the Lorenz attractor butterfly effect.

In 1961, Lorenz was running a numerical computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the full precision 0.506127 value. The result was a completely different weather scenario.

Here is the set of Lorenz equations:



Now, the set of differential equations which describe the planetary orbits is much more complicated than this.




NOTHING can be said about the RE heliocentrical system beyond a time scale of 300 YEARS.

Dr. Robert W. Bass

Ph.D. (Mathematics) Johns Hopkins University, 1955 [Wintner, Hartman]
A. Wintner, world's leading authority on celestial mechanics
Post-Doctoral Fellow Princeton University, 1955-56 [under S. Lefschetz]
Rhodes Scholar
Professor, Physics & Astronomy, Brigham Young University

"In a resonant, orbitally unstable or "wild" motion, the eccentricities of one or more of the terrestrial planets can increase in a century or two until a near collision occurs. Subsequently the Principle of Least Interaction Action predicts that the planets will rapidly "relax" into a configuration very near to a (presumably orbitally stable) resonant, Bode's-Law type of configuration. Near such a configuration, small, non-gravitational effects such as tidal friction can in a few centuries accumulate effectively to a discontinuous "jump" from the actual phase-space path to a nearby, truly orbitally stable, path. Subsequently, observations and theory would agree that the solar system is in a quasi-periodic motion stable in the sense of Laplace and orbitally stable. Also, numerical integrations backward in time would show that no near collision had ever occurred. Yet in actual fact this deduction would be false."

"I arrived independently at the preceding scenario before learning that dynamical astronomer, E. W. Brown, president of the American Astronomical Society, had already outlined the same possibility in 1931."

Dr. Robert Bass, Stability of the Solar System:

https://web.archive.org/web/20120916174745/http://www.innoventek.com:80/Bass1974PenseeAllegedProofsOfStabilityOfSolarSystemR.pdf

Dr. E.W. Brown

Fellowship, Royal Society
President of the American Mathematical Society
Professor of Mathematics, Yale University
President of the American Astronomical Society

What this means is that the interval of assured reliability for Newton's equations of gravitational motion is at most three hundred years.

Dr. W.M. Smart

Regius Professor of Astronomy at Glasgow University
President of the Royal Astronomical Society from 1949 to 1951







Within this 300 year time interval, we again have the huge problem of the sensitive dependence on initial conditions.
« Last Edit: April 27, 2019, 05:44:38 PM by sandokhan »