The Flat Earth Society
Flat Earth Discussion Boards => Flat Earth Theory => Topic started by: BRrollin on May 11, 2020, 06:53:41 PM

A fairly recent published scientific article is provided below, which analyzes possible bounded orbits using Newtonian central force in the case of 3 bodies.
The analysis demonstrates evidence that:
1. Closed bounded orbits arise from central force considerations
2. The bounded orbits identified have also been verified using numerical computations (see references therein)
3. There is a demonstrably mathematical distinction between analytical solutions, the application in numerical methods, and the application to chaotic dynamics.
https://arxiv.org/pdf/physics/0410149.pdf

A fairly recent published scientific article is provided below, which analyzes possible bounded orbits using Newtonian central force in the case of 3 bodies.
The analysis demonstrates evidence that:
1. Closed bounded orbits arise from central force considerations
2. The bounded orbits identified have also been verified using numerical computations (see references therein)
3. There is a demonstrably mathematical distinction between analytical solutions, the application in numerical methods, and the application to chaotic dynamics.
https://arxiv.org/pdf/physics/0410149.pdf
I'm very new to this site, and this seems to represent a neat opportunity for me to learn some cool physics.
I have no personal background in science, but I do love to learn  to wit, can you help elaborate, in simple terms, the "3 body problem" ? Regularly Wikipedia is too dense for me. And in reading the Wiki on this site, I'm still pretty confused about certain things. What makes it a "problem"? And what are its implications for astrophysics? And why is it such a hangup for the FET?
Here's an example of why it's hard for a laymen like me to wrap my head around: I get that it's referring to three bodies in orbit with each other, but how does that exactly matter for the solar system anyway? The moon isn't directly orbiting the sun, it's orbiting the earth, right? So, my very limited understanding of physics makes me think that in terms of gravitational forces it's really just "two" bodies we're dealing with  the sun, and the earth/moon as if it were ONE body. "3 bodies in orbit with each other" sounds like a system that has three stars all orbiting each other or something odd?
I'd love to understand  but with as little actual math as possible ;D :D

I have no personal background in science, but I do love to learn  to wit, can you help elaborate, in simple terms, the "3 body problem" ? Regularly Wikipedia is too dense for me. And in reading the Wiki on this site, I'm still pretty confused about certain things. What makes it a "problem"? And what are its implications for astrophysics? And why is it such a hangup for the FET?
Here's an example of why it's hard for a laymen like me to wrap my head around: I get that it's referring to three bodies in orbit with each other, but how does that exactly matter for the solar system anyway? The moon isn't directly orbiting the sun, it's orbiting the earth, right? So, my very limited understanding of physics makes me think that in terms of gravitational forces it's really just "two" bodies we're dealing with  the sun, and the earth/moon as if it were ONE body. "3 bodies in orbit with each other" sounds like a system that has three stars all orbiting each other or something odd?
I'd love to understand  but with as little actual math as possible ;D :D
I'm not surprised you are confused after reading the wiki here. :)
Let me try something simple. Hopefully.
Imagine a pool table. It has some billiard balls and 4 sides.
Now if you hit a ball and it strikes another ball, it's a simple calculation to predict the two balls will be going. Hit a ball straight on and it goes in the same direction. Hit a ball with a glancing blow, the balls fly off in different directions.
So anyone would agree, we know how to calculate two balls bouncing off each other. Video games do physics like this all the time. Nobody could claim we don't know how basic geometry works.
But what if we add a third ball? Now we have a problem, the equation we have is for two balls colliding. How can we possibly know how three will interact?
The solution again is simple, we just calculate pairs of billiards one at a time. Start with one, move it a tiny bit, run the equations for each of the other balls to see if there is a collision. Repeat this enough and you can simulate a full pool table.
The same with orbits. Newtons laws describe two bodies. We don't know a simple formula for 3. But we can calculate them in pairs in small steps and get answers.
Nobody says we have a "three billiard ball problem" and therefore, pool tables don't actually exist, and they are actually pooltriangles. They are real, and we can simulate them just fine. Nobody can describe a single equation that describes an entire table, but so what? Lots of problems don't have algebraic solutions, but we can solve them with other methods just fine.
Coming at it from another angle is, how did we get a probe to Pluto if we can't calculate orbits and understand how gravity behaves?

A fairly recent published scientific article is provided below, which analyzes possible bounded orbits using Newtonian central force in the case of 3 bodies.
The analysis demonstrates evidence that:
1. Closed bounded orbits arise from central force considerations
2. The bounded orbits identified have also been verified using numerical computations (see references therein)
3. There is a demonstrably mathematical distinction between analytical solutions, the application in numerical methods, and the application to chaotic dynamics.
https://arxiv.org/pdf/physics/0410149.pdf
I'm very new to this site, and this seems to represent a neat opportunity for me to learn some cool physics.
I have no personal background in science, but I do love to learn  to wit, can you help elaborate, in simple terms, the "3 body problem" ? Regularly Wikipedia is too dense for me. And in reading the Wiki on this site, I'm still pretty confused about certain things. What makes it a "problem"? And what are its implications for astrophysics? And why is it such a hangup for the FET?
Here's an example of why it's hard for a laymen like me to wrap my head around: I get that it's referring to three bodies in orbit with each other, but how does that exactly matter for the solar system anyway? The moon isn't directly orbiting the sun, it's orbiting the earth, right? So, my very limited understanding of physics makes me think that in terms of gravitational forces it's really just "two" bodies we're dealing with  the sun, and the earth/moon as if it were ONE body. "3 bodies in orbit with each other" sounds like a system that has three stars all orbiting each other or something odd?
I'd love to understand  but with as little actual math as possible ;D :D
Diddo what JSS said.
Also, when you have 3 bodies interacting gravitationally, the differential equations become tangled. So mathematicians like to worry about this and try to find mathematical solutions for them that are “analytical.” That is, the solutions are completely describable using the algebraic structures in math.
In physics, they don’t really care about that, so they find solutions that are not analytically closed, but solve the differential equations to the desired accuracy that is needed.
The reason FEers get stuck on this, and really it’s just Tom Bishop, is that in order to promote their FE idea, they want to show that modern science is somehow wrong.
Their approach here is to take the fact that since analytic solutions to the 3 body problem have not been found by mathematicians, that then Newton’s laws (which produce the equations) are wrong, and hence the fundamental basis for physics is wrong.
It really is not a problem, but FEers have a vested interest in maintaining that it somehow is.
If you can’t solve the 3 body problem, then how can you describe the solar system (which has many more bodies)?
Hope this helps. And this is all my take on it.

Thank you, both. All that REALLY helps clarify a few things.
As I wrote in other threads, I'm NOT a math guy. But I am a logic guy.
And doesn't this argument of Tom/FEers completely destroy their own FET?
Let me rephrase it to see what I mean (although I'm sure you already do, but for the sake of others who read this thread) 
The argument goes:
"Since this one thing [3 body problem] cannot be fully mathematically described, it must therefore mean physics is wrong, so we cannot rely on it to determine that gravity and the solar system operate the way science says it does."
The exact same reasoning would immediately lead anyone to conclude that FET is wrong. ??? ::) There's almost nothing in FET that is mathematically described in a consistent/coherent way. Almost everything follows a short road, then ends with "well, beyond this point we aren't sure." Examples are endless (what's the size of any celestial body we see? What's the actual path that even ONE of them take that also accounts for every single phenomena that all humans observe?).
From a nonscientist's standpoint, it seems like the more they push this argument while failing to fully mathematically describe basically all parts of their theory, the more it's clear they're missing the galaxy for the trees (so to speak ;D ).
BUT, I do have one question 
What's up with the idea that it says in the Wiki on this site that 3+ bodies become inherently unstable over time? Is that a red herring? Is it a "given zillions of years" issue? The 3 body problem section of the Wiki here devotes quite a bit of space to it, so I'd like to understand a bit more.
Thanks!!!!
(EDIT: also, and I say this with complete sincerity, I am thrilled that I am learning tidbits of actual science on a website where I expected to be informed of none).

Thank you, both. All that REALLY helps clarify a few things.
As I wrote in other threads, I'm NOT a math guy. But I am a logic guy.
And doesn't this argument of Tom/FEers completely destroy their own FET?
Let me rephrase it to see what I mean (although I'm sure you already do, but for the sake of others who read this thread) 
The argument goes:
"Since this one thing [3 body problem] cannot be fully mathematically described, it must therefore mean physics is wrong, so we cannot rely on it to determine that gravity and the solar system operate the way science says it does."
The exact same reasoning would immediately lead anyone to conclude that FET is wrong. ??? ::) There's almost nothing in FET that is mathematically described in a consistent/coherent way. Almost everything follows a short road, then ends with "well, beyond this point we aren't sure." Examples are endless (what's the size of any celestial body we see? What's the actual path that even ONE of them take that also accounts for every single phenomena that all humans observe?).
From a nonscientist's standpoint, it seems like the more they push this argument while failing to fully mathematically describe basically all parts of their theory, the more it's clear they're missing the galaxy for the trees (so to speak ;D ).
BUT, I do have one question 
What's up with the idea that it says in the Wiki on this site that 3+ bodies become inherently unstable over time? Is that a red herring? Is it a "given zillions of years" issue? The 3 body problem section of the Wiki here devotes quite a bit of space to it, so I'd like to understand a bit more.
Thanks!!!!
I get what you're saying and I've noticed that no FE idea seems to stand together/make sense with other FE ideas. For example being able to totally see that rockets aren't leaving the atmosphere but are curving off into the distance which is somehow proof that we have never gone to space, while also claiming things like extreme bendy light to explain an extremely distorted perspective view of the world or why the sun is somehow hitting near half of the world and not the other half, and yet while claiming bendy light also complain when things are explained on a round earth with mirages (like being able to see a city in the distance where you normally couldn't, in very specific conditions) "Look see, we can see that city so it proves there is no curve, bendy light is just an excuse!"
So yea, flat earth ideas/explanations (lets call them X, Y and Z) don't work together. if X, Y cant happen, if Y, Z cant happen etc etc. but they still use Y to explain a specific thing while almost intentionally staying ignorant of X and Z
I guess X Y Z is the true 3 body problem that FE have yet to solve. :P

Thank you, both. All that REALLY helps clarify a few things.
As I wrote in other threads, I'm NOT a math guy. But I am a logic guy.
And doesn't this argument of Tom/FEers completely destroy their own FET?
Let me rephrase it to see what I mean (although I'm sure you already do, but for the sake of others who read this thread) 
The argument goes:
"Since this one thing [3 body problem] cannot be fully mathematically described, it must therefore mean physics is wrong, so we cannot rely on it to determine that gravity and the solar system operate the way science says it does."
The exact same reasoning would immediately lead anyone to conclude that FET is wrong. ??? ::) There's almost nothing in FET that is mathematically described in a consistent/coherent way. Almost everything follows a short road, then ends with "well, beyond this point we aren't sure." Examples are endless (what's the size of any celestial body we see? What's the actual path that even ONE of them take that also accounts for every single phenomena that all humans observe?).
From a nonscientist's standpoint, it seems like the more they push this argument while failing to fully mathematically describe basically all parts of their theory, the more it's clear they're missing the galaxy for the trees (so to speak ;D ).
BUT, I do have one question 
What's up with the idea that it says in the Wiki on this site that 3+ bodies become inherently unstable over time? Is that a red herring? Is it a "given zillions of years" issue? The 3 body problem section of the Wiki here devotes quite a bit of space to it, so I'd like to understand a bit more.
Thanks!!!!
(EDIT: also, and I say this with complete sincerity, I am thrilled that I am learning tidbits of actual science on a website where I expected to be informed of none).
Yes, well, from my experience FEers tend to hold different standards for their own claims. I’ve witnessed a FEer criticize a piece of published RE evidence detailing valid mathematical prescriptions to a level where they pick out certain technical terms and compare them to what Poincare said, yet post a hazy video from some person of a shoreline and claim it proves the earth is flat!
So you can make your own conclusions about that.
In terms of stability, the FE argument, IMO, becomes unfocused quickly. First, there are known (and proven) stable 3body systems. So the fundamental claim they make is untrue.
However, they are correct in that most 3body systems are unstable, but they don’t seem to recognize what that means. Stability means that a perturbation from an equilibrium point will return. That’s all. So it’s not a question of whether systems are stable, but what timescales the instability will manifest observable differences.
For example, it is known that the earth is in an unstable orbit around the Sun. If you wait long enough, the Earth will spiral into the Sun. The time it will take to do this is longer than the lifetime of the Sun.
Hence, the whole focus on stability doesn’t aid the FE objective, IMO. Effectively, they are arguing a detail that never presents a problem in RE scenarios.

What's up with the idea that it says in the Wiki on this site that 3+ bodies become inherently unstable over time? Is that a red herring? Is it a "given zillions of years" issue? The 3 body problem section of the Wiki here devotes quite a bit of space to it, so I'd like to understand a bit more.
Thanks!!!!
(EDIT: also, and I say this with complete sincerity, I am thrilled that I am learning tidbits of actual science on a website where I expected to be informed of none).
I'll focus on this question and add I've learned a lot here too, both from having things explained and being forced to look stuff up in detail to try and argue.
Any complex orbital system is going to be unstable. We found some stable 3body solutions that work in pure math, but once you throw things into the real world stuff goes haywire eventually.
But yes, it's a matter of zillions of years. The moons of Jupiter and Saturn are unstable. That's why they have rings, former moons that were literally torn apart. But they won't fly apart tomorrow. But in a billion years? Sure, they will have likely changed.
In the long term, Earth could be ejected from the solar system or moved to another orbit eventually. But we call it stable because that would likely take billions or tens of billions of years. Hard to tell. But that's reality. Like with the pool table, i can't tell you exactly how a break will turn out, but if someone claims they will all end up stacked and balanced on each other up to the ceiling, i can say NO.
Just because we don't know EVERYTHING doesn't mean we don't know ANYTHING.

What's up with the idea that it says in the Wiki on this site that 3+ bodies become inherently unstable over time? Is that a red herring? Is it a "given zillions of years" issue? The 3 body problem section of the Wiki here devotes quite a bit of space to it, so I'd like to understand a bit more.
Thanks!!!!
(EDIT: also, and I say this with complete sincerity, I am thrilled that I am learning tidbits of actual science on a website where I expected to be informed of none).
I'll focus on this question and add I've learned a lot here too, both from having things explained and being forced to look stuff up in detail to try and argue.
Any complex orbital system is going to be unstable. We found some stable 3body solutions that work in pure math, but once you throw things into the real world stuff goes haywire eventually.
But yes, it's a matter of zillions of years. The moons of Jupiter and Saturn are unstable. That's why they have rings, former moons that were literally torn apart. But they won't fly apart tomorrow. But in a billion years? Sure, they will have likely changed.
In the long term, Earth could be ejected from the solar system or moved to another orbit eventually. But we call it stable because that would likely take billions or tens of billions of years. Hard to tell. But that's reality. Like with the pool table, i can't tell you exactly how a break will turn out, but if someone claims they will all end up stacked and balanced on each other up to the ceiling, i can say NO.
Just because we don't know EVERYTHING doesn't mean we don't know ANYTHING.
Yes, it is making a LOT of sense to me conceptually (which is all that can happen, because there's no way I'll understand the maths), thanks to you and BRollin.
Also, this tracks with reality, IMO. I don't know much about inertia, gravity, momentum, etc., but I can grasp them in my everyday life enough to suspect that it would NOT make sense for the solar system to NEVER become "unstable" and change. Just like a car when you take your foot off the gas, you will have inertia or momentum that carries you for some time, even on a totally flat road, but eventually other forces (gravity? friction?) work on the car and it will stop even on an endless flat road.
I have no idea if I'm super conflating something in saying this, but it's like the idea that "you can't have a perpetual motion machine" and so, the solar system, at a certain point, can't be sustained forever stably  otherwise it WOULD be a "perpetual motion machine" in the big picture. Or are those two things tooooootally different?

Yes, it is making a LOT of sense to me conceptually (which is all that can happen, because there's no way I'll understand the maths), thanks to you and BRollin.
Also, this tracks with reality, IMO. I don't know much about inertia, gravity, momentum, etc., but I can grasp them in my everyday life enough to suspect that it would NOT make sense for the solar system to NEVER become "unstable" and change. Just like a car when you take your foot off the gas, you will have inertia or momentum that carries you for some time, even on a totally flat road, but eventually other forces (gravity? friction?) work on the car and it will stop even on an endless flat road.
I have no idea if I'm super conflating something in saying this, but it's like the idea that "you can't have a perpetual motion machine" and so, the solar system, at a certain point, can't be sustained forever stably  otherwise it WOULD be a "perpetual motion machine" in the big picture. Or are those two things tooooootally different?
They are quite different actually. :)
There is no friction in space. If you had just the Earth and the Sun, the Earth would continue orbiting forever with no changes. It takes no power to keep Earth in it's orbit, as far as the Earth is concerned it's moving in a straight line. It's just that line is bent into a closed circle by the Suns gravity. It's not a "perpetual motion machine" because it doesn't produce or consume energy, it is just stable.
With multiple bodies it's more complex, but energy is still conserved. If a small asteroid gets too close to Jupiter, it can get pulled in and flung out at high speeds, but Jupiter will also lose a tiny fraction of it's momentum. So you have one object going faster and one going slower, but that asteroid could loop back and hit Jupiter and give it's kinetic energy back too. So humans can use Jupiter to fling spacecraft into deep space and it is effectively an infinite source of gravitational energy, but if we used it enough we could drop Jupiter into the sun by stealing it all. If we keep doing it for, oh, a trillion years.
So the Earth won't stop, but if other bodies get close it can be nudged into other orbits, possibly out of the solar system or into the sun. But not for a LOOOOOOONG time, If ever.

Yes, it is making a LOT of sense to me conceptually (which is all that can happen, because there's no way I'll understand the maths), thanks to you and BRollin.
Also, this tracks with reality, IMO. I don't know much about inertia, gravity, momentum, etc., but I can grasp them in my everyday life enough to suspect that it would NOT make sense for the solar system to NEVER become "unstable" and change. Just like a car when you take your foot off the gas, you will have inertia or momentum that carries you for some time, even on a totally flat road, but eventually other forces (gravity? friction?) work on the car and it will stop even on an endless flat road.
I have no idea if I'm super conflating something in saying this, but it's like the idea that "you can't have a perpetual motion machine" and so, the solar system, at a certain point, can't be sustained forever stably  otherwise it WOULD be a "perpetual motion machine" in the big picture. Or are those two things tooooootally different?
They are quite different actually. :)
There is no friction in space. If you had just the Earth and the Sun, the Earth would continue orbiting forever with no changes. It takes no power to keep Earth in it's orbit, as far as the Earth is concerned it's moving in a straight line. It's just that line is bent into a closed circle by the Suns gravity. It's not a "perpetual motion machine" because it doesn't produce or consume energy, it is just stable.
With multiple bodies it's more complex, but energy is still conserved. If a small asteroid gets too close to Jupiter, it can get pulled in and flung out at high speeds, but Jupiter will also lose a tiny fraction of it's momentum. So you have one object going faster and one going slower, but that asteroid could loop back and hit Jupiter and give it's kinetic energy back too. So humans can use Jupiter to fling spacecraft into deep space and it is effectively an infinite source of gravitational energy, but if we used it enough we could drop Jupiter into the sun by stealing it all. If we keep doing it for, oh, a trillion years.
So the Earth won't stop, but if other bodies get close it can be nudged into other orbits, possibly out of the solar system or into the sun. But not for a LOOOOOOONG time, If ever.
Once again, so good at explaining! Okay, so, after the first sentence that there's no friction in space, my brain went "oh, cool, so maybe we COULD have a perpetual motion machine...IN SPACE? Friction is the problem!" But after reading everything, this is not the case.
I do love the idea that an alien civilization with a timespan measured in trillions of years, can accidentally come to the point of sending a planet in their solar system out of orbit by using it countless times to fling spacecraft into deeper space, thereby reducing its kinetic(?) energy until it can't sustain its orbit. by the way, what is actually happening there, in terms of its change in course? Is its orbit getting infinitesimally smaller? Or bigger? Or it depends?

Once again, so good at explaining! Okay, so, after the first sentence that there's no friction in space, my brain went "oh, cool, so maybe we COULD have a perpetual motion machine...IN SPACE? Friction is the problem!" But after reading everything, this is not the case.
I do love the idea that an alien civilization with a timespan measured in trillions of years, can accidentally come to the point of sending a planet in their solar system out of orbit by using it countless times to fling spacecraft into deeper space, thereby reducing its kinetic(?) energy until it can't sustain its orbit. by the way, what is actually happening there, in terms of its change in course? Is its orbit getting infinitesimally smaller? Or bigger? Or it depends?
Thanks, I do try.
An orbit will get larger if you add energy to it, and smaller if you take it away. So if Jupiter looses a bit of speed with each spacecraft boost, it will eventually fall into the sun.
I may have vastly underestimated the amount of time it would take to do this however. Reading up, the Voyager 1 spacecraft stole enough speed from Jupiter to slow the planet down by one foot per trillion years. So a trillion years from now, it will be one foot behind in it's orbit. That's not much!
To drop it into the Sun that way would likely take longer than I care to calculate, by a lot. Of course, if they start launching small moons, that would speed things up.

Once again, so good at explaining! Okay, so, after the first sentence that there's no friction in space, my brain went "oh, cool, so maybe we COULD have a perpetual motion machine...IN SPACE? Friction is the problem!" But after reading everything, this is not the case.
I do love the idea that an alien civilization with a timespan measured in trillions of years, can accidentally come to the point of sending a planet in their solar system out of orbit by using it countless times to fling spacecraft into deeper space, thereby reducing its kinetic(?) energy until it can't sustain its orbit. by the way, what is actually happening there, in terms of its change in course? Is its orbit getting infinitesimally smaller? Or bigger? Or it depends?
Thanks, I do try.
An orbit will get larger if you add energy to it, and smaller if you take it away. So if Jupiter looses a bit of speed with each spacecraft boost, it will eventually fall into the sun.
I may have vastly underestimated the amount of time it would take to do this however. Reading up, the Voyager 1 spacecraft stole enough speed from Jupiter to slow the planet down by one foot per trillion years. So a trillion years from now, it will be one foot behind in it's orbit. That's not much!
To drop it into the Sun that way would likely take longer than I care to calculate, by a lot. Of course, if they start launching small moons, that would speed things up.
Ah, yes. So, as with lots of things in the universe, it's really hard to grasp. A trillion years is like two hundred times longer than the age of the earth, so it's functionally the same as saying "Jupiter will NEVER fall into the sun." It goes to the same thing you or someone said earlier (I think) that the sun will die before the earth's orbit will decay into it.

All sources on this subject admit that they failed at predicting the motion of celestial systems past two bodies. Can you point me to the part in history where scientists were able to use the three and nbody problems to describe the solar system based on Newton's laws? Surely this would have been mentioned somewhere in conventional knowledge materials.
https://youtu.be/fUsePzlOmxw?t=581

All sources on this subject admit that they failed at predicting the motion of celestial system past two bodies. Can you point me to the part in history where scientists were able to use the three and nbody problems to describe the solar system based on Newton's laws? Surely this would have been mentioned somewhere in conventional knowledge materials.
https://youtu.be/fUsePzlOmxw?t=581
Can you point to any credible scientists today that have said that nbody problems means the world isn't spherical?

All sources on this subject admit that they failed at predicting the motion of celestial system past two bodies. Can you point me to the part in history where scientists were able to use the three and nbody problems to describe the solar system based on Newton's laws? Surely this would have been mentioned somewhere in conventional knowledge materials.
I have no idea how you can keep repeating this. What sources say we can't predict the motion of celestial systems past two bodies? Nobody. All you can find are math papers saying we can't solve the problems using ONE method, but there are thousands talking about other methods that work just fine. Plus we have all those spacecraft zooming all over that seem to find their way, and the oftmentioned comets hitting Jupiter that were predicted perfectly.
No matter how many times you say we can't, predictions prove you are wrong.

All sources on this subject admit that they failed at predicting the motion of celestial system past two bodies. Can you point me to the part in history where scientists were able to use the three and nbody problems to describe the solar system based on Newton's laws? Surely this would have been mentioned somewhere in conventional knowledge materials.
I have no idea how you can keep repeating this. What sources say we can't predict the motion of celestial systems past two bodies? Nobody. All you can find are math papers saying we can't solve the problems using ONE method, but there are thousands talking about other methods that work just fine. Plus we have all those spacecraft zooming all over that seem to find their way, and the oftmentioned comets hitting Jupiter that were predicted perfectly.
No matter how many times you say we can't, predictions prove you are wrong.
Are you going to give us a source explaining how physcists were able to overcome the Three Body Problem to describe the SunEarthMoon system, in contradiction to the Nova documentary's contrary statements of what happened after Newton published his laws? Or will you continue to cite your own self?
The basic scheme of RE cosmology says that it is possible to have a star with a planet and a moon. Hundreds of years of research by the greatest mathematicians have been unable to get that to work, however.
It is pretty damning that the basic idea of a star with a planet and a moon can't stay together and that the accepted model does not have working laws.

All sources on this subject admit that they failed at predicting the motion of celestial system past two bodies. Can you point me to the part in history where scientists were able to use the three and nbody problems to describe the solar system based on Newton's laws? Surely this would have been mentioned somewhere in conventional knowledge materials.
I have no idea how you can keep repeating this. What sources say we can't predict the motion of celestial systems past two bodies? Nobody. All you can find are math papers saying we can't solve the problems using ONE method, but there are thousands talking about other methods that work just fine. Plus we have all those spacecraft zooming all over that seem to find their way, and the oftmentioned comets hitting Jupiter that were predicted perfectly.
No matter how many times you say we can't, predictions prove you are wrong.
Are you going to give us a source explaining how physcists were able to overcome the Three Body Problem to describe the SunEarthMoon system, in contradiction to the Nova documentary's contrary statements of what happened after Newton published his laws? Or will you continue to cite your own self?
The basic scheme of RE cosmology says that it is possible to have a star with a planet and a moon. Hundreds of years of research by the greatest mathematicians have been unable to get that to work, however.
Here's an interesting paper, "MoonEarthSun: The oldest threebody problem", that discusses all of the solutions from Kepler on up through Newton to roughly today. A lot of stuff has happened since Newton. All with ever increasing accuracy. The bottom line is that for our purposes we can accurately predict the MoonEarthSun system movement to a high degree of precision.
http://sites.apam.columbia.edu/courses/ap1601y/MoonEarthSin%20RMP.70.589.pdf
It is pretty damning that the basic idea of a star with a planet and a moon can't stay together and that the accepted model does not have working laws.
It's more damning that FET has no knowledge of where a star and a moon are in relation to a planet at all. Helio can give a pretty precise approximation and prediction as evidenced by some of the work referenced in the paper above. FET has no knowledge of any celestial mechanics. Zero. FET doesn't know where any objects in the heavens are or their size let alone being able to predict anything. Your time may be better spent trying to show how the Sun, Moon, and planets, even comets, work within FET rather than looking for holes in Newton.

All sources on this subject admit that they failed at predicting the motion of celestial system past two bodies. Can you point me to the part in history where scientists were able to use the three and nbody problems to describe the solar system based on Newton's laws? Surely this would have been mentioned somewhere in conventional knowledge materials.
I have no idea how you can keep repeating this. What sources say we can't predict the motion of celestial systems past two bodies? Nobody. All you can find are math papers saying we can't solve the problems using ONE method, but there are thousands talking about other methods that work just fine. Plus we have all those spacecraft zooming all over that seem to find their way, and the oftmentioned comets hitting Jupiter that were predicted perfectly.
No matter how many times you say we can't, predictions prove you are wrong.
Are you going to give us a source explaining how physcists were able to overcome the Three Body Problem to describe the SunEarthMoon system, in contradiction to the Nova documentary's contrary statements of what happened after Newton published his laws? Or will you continue to cite your own self?
The basic scheme of RE cosmology says that it is possible to have a star with a planet and a moon. Hundreds of years of research by the greatest mathematicians have been unable to get that to work, however.
It is pretty damning that the basic idea of a star with a planet and a moon can't stay together and that the accepted model does not have working laws.
Round and round we go.
We predicted Comet Shoemaker–Levy's multiple impacts with Jupiter over a year in advance using Newton's math. Explain how we did that if the math doesn't work?

Here's an interesting paper, "MoonEarthSun: The oldest threebody problem", that discusses all of the solutions from Kepler on up through Newton to roughly today. A lot of stuff has happened since Newton. All with ever increasing accuracy. The bottom line is that for our purposes we can accurately predict the MoonEarthSun system movement to a high degree of precision.
http://sites.apam.columbia.edu/courses/ap1601y/MoonEarthSin%20RMP.70.589.pdf
Funny, the author of that paper concludes in the ending section of that paper that Newton's laws are not a sufficient explanation:
Many physicists may be tempted to see in Newton’s equations of motion and his universal gravitation a sufficient explanation for the threebody problem, with the details to be worked out by the technicians. But even a close look at the differential equations (29) and (30) does not prepare us for the idiosyncracies of the lunar motion, nor does it help us to understand the orbits of asteroids in the combined gravitational field of the Sun and Jupiter.
We predicted Comet Shoemaker–Levy's multiple impacts with Jupiter over a year in advance using Newton's math. Explain how we did that if the math doesn't work?
That was discussed in the other thread on that matter. They only predicted a portion of an orbit, and are using other methods such as epicycles.

We predicted Comet Shoemaker–Levy's multiple impacts with Jupiter over a year in advance using Newton's math. Explain how we did that if the math doesn't work?
That was discussed in the other thread on that matter. They only predicted a portion of an orbit, and are using other methods such as epicycles.
Yes it was, and it was pointed out they predicted nearly a full orbit, and papers were linked that clearly stated they used Newtons methods not 'epicycles'.
Epicycles wouldn't work anyway as the final orbit was bent inward toward Jupiter as it got closer.
The simple fact is the math WORKS. I have to believe actual predictions over one persons insistence that it doesn't.

We predicted Comet Shoemaker–Levy's multiple impacts with Jupiter over a year in advance using Newton's math. Explain how we did that if the math doesn't work?
That was discussed in the other thread on that matter. They only predicted a portion of an orbit, and are using other methods such as epicycles.
Yes it was, and it was pointed out they predicted nearly a full orbit, and papers were linked that clearly stated they used Newtons methods not 'epicycles'.
Epicycles wouldn't work anyway as the final orbit was bent inward toward Jupiter as it got closer.
The simple fact is the math WORKS. I have to believe actual predictions over one persons insistence that it doesn't.
No, those papers said that they were using perturbation methods, which are epicycles which use a two body problem as the underlying model instead of Ptolmy's circle. You have not yet provided a rebuttal with appropriate sources to the sources given explaining this.

We predicted Comet Shoemaker–Levy's multiple impacts with Jupiter over a year in advance using Newton's math. Explain how we did that if the math doesn't work?
That was discussed in the other thread on that matter. They only predicted a portion of an orbit, and are using other methods such as epicycles.
Yes it was, and it was pointed out they predicted nearly a full orbit, and papers were linked that clearly stated they used Newtons methods not 'epicycles'.
Epicycles wouldn't work anyway as the final orbit was bent inward toward Jupiter as it got closer.
The simple fact is the math WORKS. I have to believe actual predictions over one persons insistence that it doesn't.
No, those papers said that they were using perturbation methods, which are epicycles which use a two body problem as the underlying model instead of Ptolmy's circle. You have not yet provided a rebuttal with appropriate sources to the sources given explaining this.
Yes, they were using models based on Newtons 2body equations. Which is exactly how one uses Newton's laws to calculate orbits. Perturbation methods are a very well known and often used method for solving complex math. Especially with the advent of computers that allows huge numbers of calculations to be run to as much precision as you need.
https://en.wikipedia.org/wiki/Perturbation_theory
So as you stated, they used Newtons 2body formulas combined with Perturbation theory to correctly predict a series of comet collisions. Seems like Newton's math works.

From your article we see:
Perturbation theory was first devised to solve otherwise intractable problems (https://en.wikipedia.org/wiki/Threebody_problem) (links us to the Three Body Problem page) in the calculation of the motions of planets in the solar system. For instance, Newton's law of universal gravitation explained the gravitation between two astronomical bodies, but when a third body is added, the problem was, "How does each body pull on each?" Newton's equation only allowed the mass of two bodies to be analyzed. The gradually increasing accuracy of astronomical observations led to incremental demands in the accuracy of solutions to Newton's gravitational equations, which led several notable 18th and 19th century mathematicians, such as Lagrange and Laplace, to extend and generalize the methods of perturbation theory.
From the same article, see bolded:
Perturbation theory is closely related to methods used in numerical analysis. The earliest use of what would now be called perturbation theory was to deal with the otherwise unsolvable mathematical problems of celestial mechanics: for example the orbit of the Moon, which moves noticeably differently from a simple Keplerian ellipse because of the competing gravitation of the Earth and the Sun.[2]
Perturbation methods start with a simplified form of the original problem, which is simple enough to be solved exactly. In celestial mechanics, this is usually a Keplerian ellipse. Under nonrelativistic gravity, an ellipse is exactly correct when there are only two gravitating bodies (say, the Earth and the Moon) but not quite correct when there are three or more objects (say, the Earth, Moon, Sun, and the rest of the solar system) and not quite correct when the gravitational interaction is stated using formulations from General relativity.
It's a work around for otherwise unsolvable mathematical problems of celestial mechanics.
How is it that they can't simulate three bodies, but they have a work around which involves computing the effects bodies have on each other?
Physcist Dr. Gopi Krishna Vijaya says that, in using Perturbation Theory, astronomers are really using epicycles with a gravitational disguise.
Replacing the Foundations of Astronomy  .pdf (https://archive.org/download/replacingthefoundationsofastronomyvijayagopikrishna2/Replacing%20the%20Foundations%20of%20Astronomy%20%28Vijaya%2C%20Gopi%20Krishna%29%20%282%29.pdf)
Epicycles Once More
“ Following the Newtonian era, in the 18th century there were a series of mathematicians – Bernoulli, Clairaut, Euler, D’Alembert, Lagrange, Laplace, Leverrier – who basically picked up where Newton left off and ran with it. There were no descendants to the wholistic viewpoints of Tycho and Kepler, but only those who made several improvements of a mathematical nature to Newtonian theory. Calculus became a powerful tool in calculating the effects of gravitation of all the planets upon each other, due to their assumed masses. The motion of the nearest neighbor – the Moon – was a surprisingly hard nut to crack even for Newton, and several new mathematical techniques had to be invented just to tackle that.
In the process, a new form of theory became popular: Perturbation theory. In this approach, a small approximate deviation from Newton's law is assumed, based on empirical data, and then a rigorous calculation of differential equation is used to nail down the actual value of the deviation. It does not take much to recognize that this was simply the approach taken before Kepler by Copernicus and others for over a thousand years – adding epicycles to make the observations fit. It is the same concept, but now dressed up in gravitational disguise: ”
(https://i.imgur.com/KiTaMfy.png)
“ In other words, the entire thought process took several steps backwards, to redo the same process as the Ptolemaic  Copernican epicycle theory, only with different variables. The more logical way of approach would have been to redirect the focus of the improved mathematical techniques to the assumptions in Newton’s theory, but instead the same equations were rederived with calculus, without examining the assumptions. Hence any modern day textbook gives the same derivation for circular and elliptical motion that Newton first derived in his Principia. The equivalence of the epicycle theory and gravitational theory has not been realized, and any new discovery that fits in with the mathematical framework of Newtonian gravity is lauded as a “triumph of the theory of gravitation.” In reality, it is simply the triumph of fitting curves to the data or minor linear extrapolations – something that had already been done at least since 2nd century AD. Yet the situation is conceptually identical. ”
Celestial Mechanics Professor Charles Lane Poor said (https://archive.org/stream/gravitationvers00chamgoog#page/n174/mode/2up):
The deviations from the “ideal” in the elements of a planet’s orbit are called “perturbations” or “variations”.... In calculating the perturbations, the mathematician is forced to adopt the old device of Hipparchus, the discredited and discarded epicycle. It is true that the name, epicycle, is no longer used, and that one may hunt in vain through astronomical textbooks for the slightest hint of the present day use of this device, which in the popular mind is connected with absurd and fantastic theories. The physicist and the mathematician now speak of harmonic motion, of Fourier’s series, of the development of a function into a series of sines and cosines. The name has been changed, but the essentials of the device remain. And the essential, the fundamental point of the device, under whatever name it may be concealed, is the representation of an irregular motion as the combination of a number of simple, uniform circular motions.
We saw earlier from your source that the actual way, The Three Body Problem, the real dynamical system under Newton's laws, doesn't work.

It's a work around for otherwise unsolvable mathematical problems of celestial mechanics.
How is it that they can't simulate three bodies, but they can have this work around to accurately compute the bodies?
Easily. When you can't solve an algebraic set of equations, you use other methods to approximate to however high a degree as you need. Numeric integration is a commonly used method in physics and math in general. You can't use PI directly in the real world, but you can use a numeric approximation.
Numerical analysis continues this long tradition: rather than exact symbolic answers, which can only be applied to realworld measurements by translation into digits, it gives approximate solutions within specified error bounds.  https://en.wikipedia.org/wiki/Numerical_analysis

Ptolmy used numerical procedures in the Almagest too. What makes you think that these numerical procedures using sines and cosines and fourier methods for perturbation analysis aren't talking about epicycles?
https://books.google.com/books?id=JVhTtVA2zr8C&pg=PA29&source=gbs_toc_r&cad=4#v=onepage&q&f=false
(https://i.imgur.com/tLawyDV.png)

Ptolmy used numerical procedures in the Almagest too. What makes you think that these numerical procedures using sines and cosines and fourier methods for perturbation analysis aren't talking about epicycles?
https://books.google.com/books?id=JVhTtVA2zr8C&pg=PA29&source=gbs_toc_r&cad=4#v=onepage&q&f=false
(https://i.imgur.com/tLawyDV.png)
Yes, numerical methods are a very old technique. There is nothing wrong with using it, it's how we convert pure math into actual numbers without solving exact equations.
The numerical point of view goes back to the earliest mathematical writings. A tablet from the Yale Babylonian Collection (YBC 7289), gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square.  https://en.wikipedia.org/wiki/Numerical_analysis
From https://en.wikipedia.org/wiki/Deferent_and_epicycle ...
Newtonian or classical mechanics eliminated the need for deferent/epicycle methods altogether and produced more accurate theories. By treating the Sun and planets as point masses and using Newton's law of universal gravitation, equations of motion were derived that could be solved by various means to compute predictions of planetary orbital velocities and positions. Simple twobody problems, for example, can be solved analytically. Morecomplex nbody problems require numerical methods for solution.
The power of Newtonian mechanics to solve problems in orbital mechanics is illustrated by the discovery of Neptune. Analysis of observed perturbations in the orbit of Uranus produced estimates of the suspected planet's position within a degree of where it was found. This could not have been accomplished with deferent/epicycle methods.
That said, epicycles are not some sort of poision. If an astronomer uses one it's because he's decided it's good enough for the result he needs. We have a large numbers of mathematical tools at our disposal, and use whichever best fits our needs. Be it accuracy or speed or ease of understanding the problem.

Perturbation Theory not really that legitimate, and is the backwards way of doing science by starting with the solution and building corrections from the ideal state (like Ptolmy did with his epicycles  he started with the observation and his "perfect" model and built corrections to match it with epicycles). It says as much on the Discovery of Neptune page: https://en.wikipedia.org/wiki/Discovery_of_Neptune
Adams learned of the irregularities while still an undergraduate and became convinced of the "perturbation" hypothesis.
...In modern terms, the problem is an inverse problem, an attempt to deduce the parameters of a mathematical model from observed data.
This is, of course, exactly opposite of using a mathematical model to predict.
That is what Dr. Gopi Krishna Vijaya explained what was happening. They 'fit' data to observations with a series of corrections.
The Wikipedia article on Perturbation Theory:
This general procedure is a widely used mathematical tool in advanced sciences and engineering: start with a simplified problem and gradually add corrections that make the formula that the corrected problem becomes a closer and closer match to the original formula.
The book Approximate Analytical Methods for Solving Ordinary Differential Equations states on p.65 (https://books.google.com/books?id=wibcBQAAQBAJ&lpg=PA65&pg=PA65#v=onepage&q&f=false):
The perturbation theory had its roots in early studies of celestial mechanics, for which the theory of epicycles was used to make small corrections to the prediction of the path of planets. Later, Charles Eugene used it to study the solar system, in particular the earthsun moon system. Now, it finds applications in many fields, such as fluid dynamics, quantum mechanics, quantum chemistry, quantum field theory, and so on.
The idea behind the perturbation method is that we start with a simplified form of the original problem (which may be extremely difficult to handle) and then gradually add corrections or perturbations, so that the solution matches that of the original problem. The simplified form is obtained by letting the perturbation parameter take the zero value.
What they were doing is mapping out all of the corrections that they had to make to Uranus and deducing that "these corrections are cause by Jupiter" and "these corrections are from x," etc. The models used tend to have hundreds and thousands of corrections/perturbations. They found odd a buildup irregularities from the corrections and concluded that it was from another planet.

Also, the claimed Discovery of Neptune success of Perturbation Theory was discredited by some, who declared that they had only discovered Neptune by luck.
From Earthsky.org  https://earthsky.org/humanworld/todayinsciencediscoveryofneptune
Ironically, as it turns out, both Le Verrier and Adams had been very lucky. Their predictions indicated Neptune’s distance correctly around 18401850. Had they made their calculations at another time, both predicted positions would have been off. Their calculations would have predicted the planet’s position only 165 years later or earlier, since Neptune takes 165 years to orbit once around the sun.
By the way, Neptune might have been discovered without the aid of mathematics. Like all planets in our solar system – because it’s closer to us than the stars – it can be seen from Earth to move apart from the star background. For example, the great astronomer Galileo, using one of the first telescopes, is said to have recorded Neptune as a faint star in 1612. If it had watched it over several weeks, he’d have noticed its unusual motion.
http://www.helas.gr/conf/2011/posters/S_5/dallas.pdf
Airy seems to be the only scientist involved in the discovery that has thoughts of a possible modification of Newtonian gravity to explain the irregular movement of Uranus. But nowhere in his memoire is there a statement that the discovery of Neptune is a test, let alone a critical one, of the law of gravitation. It was apparent shortly after the discovery that luck played its part in the easy discovery of Neptune. The whole process is extremely error prone, in both the calculations and the observations, so if the planet were not discovered in the circumstances of 1846, this would not be a refutation of Newtonian gravity, but simply a refutation of the auxiliary prepositions.
See the following from astronomer Sears C. Walker (https://books.google.com/books?id=ONX3Pr1MD8C&pg=PA134&lpg=PA134#v=onepage&q&f=false):
If we admit for the moment that my views are correct, then LeVerrier's announcement of March 29th is in perfect accordance with that of Professor Peirce of the 16th of the same month, viz. that the present visible planet Neptune is not the mathematical planet to which theory had directed the telescope. None of its elements conform to the theoretical limits. Nor does it perform the functions on which alone its existence was predicted, viz. those of removing that opprobrium of astronomers, the unexplained perturbations of Uranus.
We have it on the authority of Professor Peirce that if we ascribe to Neptune a mass of threefourths of the amount predicted by LeVerrier, it will have the best possible effect in reducing the residual perturbations of Uranus below their former value; but will nevertheless leave them on the average twothirds as great as before.
It is indeed remarkable that the two distinguished European astronomers, LeVerrier and Adams, should, by a wrong hypothesis, have been led to a right conclusion respecting the actual position of a planet in the heavens. It required for their success a compensation of errors. The unforeseen error of sixty years in their assumed period was compensated by the other unforeseen error of their assumed office of the planet. If both of them had committed only one theoretical error, (not then, but now believed to be such,) they would, according to Prof. Peirce's computations, have agreed in pointing the telescope in the wrong direction, and Neptune might have been unknown for years to come.

What you posted about Neptune verifies that Perturbation Theory not legitimate, and is the backwards way of doing science by starting with the solution and building corrections from the ideal state.
It says as much on the Discovery of Neptune page: https://en.wikipedia.org/wiki/Discovery_of_Neptune
In modern terms, the problem is an inverse problem, an attempt to deduce the parameters of a mathematical model from observed data.
What they were doing is mapping out all of the corrections that they had to make to Uranus and guessing that "these epicycles are cause by Jupiter" and "these epicycles are from x," etc. The models used have thousands of perturbations (epicycles).
How is that backwards? They had observations that included anomalies, came up with a hypothesis that something big must be out there causing them, turned it into a theory that predicted where it must be and then found it. Classic science. They also got lucky, also well established in science. But that luck only worked because they did know where to look, just not when. The orbit was still correct.
Further, the success of Perturbation Theory and Gravitation was later discredited.
...
So not everyone agreed that this claim is a proof of anything.
As many things are, we can argue this one round and round. Nothing in any of those quotes "discredits Perturbation Theory and Gravitation". At most it throws doubt onto how accurate the discovery of Neptune was. It certainly doesn't suggest as you do that it somehow totally invalidates Perturbation Theory. That's an entire branch of math which orbital mechanics is just one small use case, and thousands of mathematicians would want to have words with you if you tried to claim it was 'discredited'.
Even your quote says as much  The whole process is extremely error prone, in both the calculations and the observations, so if the planet were not discovered in the circumstances of 1846, this would not be a refutation of Newtonian gravity, but simply a refutation of the auxiliary prepositions
Perturbation Theory is extremely well established math, and none of those articles even hints it's even slightly wrong.
Read the entire wiki page about it, nowhere does it say it's discredited  https://en.wikipedia.org/wiki/Perturbation_theory

The above quotes state that Neptune was discovered by luck, rather than the accuracy or reliability of the mathematical model.
Science historian Nicholas Kollerstrom also states that dishonesty was involved with this claim.
https://en.m.wikipedia.org/wiki/Discovery_of_Neptune
In an interview in 2003, historian Nicholas Kollerstrom concluded that Adams's claim to Neptune was far weaker than had been suggested, as he had vacillated repeatedly over the planet's exact location, with estimates ranging across 20 degrees of arc. Airy's role as the hidebound superior willfully ignoring the upstart young intellect was, according to Kollerstrom, largely constructed after the planet was found, in order to boost Adams's, and therefore Britain's, credit for the discovery.
The planets are all following a path called the ecliptic, and aren't randomly distributed in the sky. 20 degrees of arc? Really?
LeVerrier later went on to "discover" the planet Vulcan with his same perturbation methods.  http://adsabs.harvard.edu/full/1953ASPL....6..291E
One of the most intemfing stories in astronomy
concerns the history of what was once believed
to be an intramercurial planet—Vulcan. The story
begins in 1859 when the planet was “discovered’
by Urbain Jean Joseph Leverrier.
Leverrier was born on March 11, 1811, the son
of a French civil servant. His birthplace, Saint
L6, is now familiar to most of us as the place
where the allies “broke out” after the Normandy
landings in the last world war.
Jean Leverrier introduced a new concept into
planetary discovery. As he was a mathematical,
rather than an observational astronomer, there is
some basis for the belief that he never looked
through a telescope. He discovered planets while
seated at his desk in the Ecole Polytechnique in
Paris. His first discovery was Neptune; his second
was Vulcan.
We all know how that one turned out.

The above quotes state that Neptune was discovered by luck, rather than the accuracy or reliability of the mathematical model.
The planets are all following a path called the Ecliptic, and aren't randomly distributed in the sky. This isn't some amazing thing.
Also, LeVerrier later went on to "discover" the planet Vulcan with his same perturbation methods.  http://adsabs.harvard.edu/full/1953ASPL....6..291E
The planet Vulcan is interesting. Yet again showing how well science works. People noticed Mercury was slightly off in it's orbit and there were numerous theories proposed. One of these was suggesting a planet Vulcan could be causing it in the 1850's. Then another theory proposed a solution half a century later, Einstein's theory of relativity. This did fit the observations perfectly, and thus once again science worked. Several theories were developed and one was eventually shown to be correct.
Now that WAS down purely to the reliability of the mathematical model. Einsteins theory is so well tested and successful that we have yet to prove it off in any way. Every test, even measuring how spacetime twists around rotating objects has been tested and matches just what we would expect.
Vulcan's a great success story.

The above quotes state that Neptune was discovered by luck, rather than the accuracy or reliability of the mathematical model.
The planets are all following a path called the Ecliptic, and aren't randomly distributed in the sky. This isn't some amazing thing.
Also, LeVerrier later went on to "discover" the planet Vulcan with his same perturbation methods.  http://adsabs.harvard.edu/full/1953ASPL....6..291E
The planet Vulcan is interesting. Yet again showing how well science works. People noticed Mercury was slightly off in it's orbit and there were numerous theories proposed. One of these was suggesting a planet Vulcan could be causing it in the 1850's. Then another theory proposed a solution half a century later, Einstein's theory of relativity. This did fit the observations perfectly, and thus once again science worked. Several theories were developed and one was eventually shown to be correct.
Now that WAS down purely to the reliability of the mathematical model. Einsteins theory is so well tested and successful that we have yet to prove it off in any way. Every test, even measuring how spacetime twists around rotating objects has been tested and matches just what we would expect.
Vulcan's a great success story.
Quite the contrary, it shows that Perturbation Theory predictions are really a guessing game. Hypothesizing the existence of undiscovered planets based on perceived irregularities rather than mathematical certainty.
As for Einstein, not everyone agreed with that one either:
Relativity and the Motion of Mercury
Charles Lane Poor, Ph.D.
Professor Emeritus of Celestial Mechanics,
Columbia University
Link to Paper (http://www.gsjournal.net/ScienceJournals/Historical%20PapersAstrophysics/Download/3394). From the Introduction:
“ Does the relativity theory, as asserted by Einstein, explain and account for even the single motion of tile perihelion of Mercury? In what way do the formulas of relativity differ from those of the classical mathematics of Newton, and how do these new formulas explain this motion? It is the purpose of this paper to discuss this single phase of the matter; to show that the very equations, or formulas, cited by the relativists as furnishing an explanation of this motion, utterly fail to furnish such an explanation. The formulas of relativity dynamics can not and do not explain the observed perihelial motion of Mercury. ”
The Theory of Mercury’s Anomalous Precession
Roger A. Rydin, Sc.D.
Associate Professor Emeritus of Nuclear Engineering,
University of Virginia
Link to Paper (http://www.tychos.info/citation/126A_MercuryPrecession.pdf)
Abstract: “ Urbain Le Verrier published a preliminary paper in 1841 on the Theory of Mercury, and a definitive paper in 1859. He discovered a small unexplained shift in the perihelion of Mercury of 39” per century. The results were corrected in 1895 by Simon Newcomb, who increased the anomalous shift by about 10%. Albert Einstein, at the end of his 1916 paper on General Relativity, gave a specific solution for the perihelion shift which exactly matched the discrepancy. Dating from the 1947 Clemence review paper, that explanation and precise value have remained to the present time, being completely accepted by theoretical physicists as absolutely true. Modern numerical fittings of planetary orbits called Ephemerides contain linearized General Relativity corrections that cannot be turned off to see if discrepancies between observation and computation still exist of the magnitude necessary to support the General Relativity estimates of the differences.
The highly technical 1859 Le Verrier paper was written in French. The partial translation given here throws light on Le Verrier’s analysis and thought processes, and points out that the masses he used for Earth and Mercury are quite different from present day values. A 1924 paper by a professor of Celestial Mechanics critiques both the Einstein and the Le Verrier analyses, and a 1993 paper gives a different and better fit to some of Le Verrier’s data. Nonetheless, the effect of errors in planet masses seems to give new condition equations that do not change the perihelion discrepancy by a large amount. The question now is whether or not the excess shift of the perihelion of Mercury is real and has been properly explained in terms of General Relativity, or if there are other reasons for the observations. There are significant arguments that General Relativity has not been proven experimentally, and that it contains mathematical errors that invalidate its predictions. Vankov has analyzed Einstein’s 1915 derivation and concludes that when an inconsistency is corrected, there is no perihelion shift at all! ”
Einstein’s Explanation of Perihelion Motion of Mercury
Hua Di
Academician, Russian Academy of Cosmonautics
Research Fellow (ret.), Stanford University
From Unsolved Problems in Special and General Relativity (https://web.archive.org/web/20190626004144/https://pdfs.semanticscholar.org/1d79/77c98a75068a913707aaec41b31ad967d562.pdf)
p.7
“ Einstein’s general theory of relativity cannot explain Mercury’s perihelion motion. He obtained “for the planet Mercury, a perihelion advance of 43” per century” by an incorrect integral calculus and many arbitrary approximations. His formula (1) is a poorly patched wrong result, tailored specially for Mercury. That is why his formula (1) fails to explain the perihelion motions for Earth and Mars. Einstein was unfair to blame “the small eccentricities of the orbits of these planets” for his failure. To sum up, Einstein’s general theory of relativity is dubious. ”

The above quotes state that Neptune was discovered by luck, rather than the accuracy or reliability of the mathematical model.
The planets are all following a path called the Ecliptic, and aren't randomly distributed in the sky. This isn't some amazing thing.
Also, LeVerrier later went on to "discover" the planet Vulcan with his same perturbation methods.  http://adsabs.harvard.edu/full/1953ASPL....6..291E
The planet Vulcan is interesting. Yet again showing how well science works. People noticed Mercury was slightly off in it's orbit and there were numerous theories proposed. One of these was suggesting a planet Vulcan could be causing it in the 1850's. Then another theory proposed a solution half a century later, Einstein's theory of relativity. This did fit the observations perfectly, and thus once again science worked. Several theories were developed and one was eventually shown to be correct.
Now that WAS down purely to the reliability of the mathematical model. Einsteins theory is so well tested and successful that we have yet to prove it off in any way. Every test, even measuring how spacetime twists around rotating objects has been tested and matches just what we would expect.
Vulcan's a great success story.
Quite the contrary, it shows that Perturbation Theory predictions are really a guessing game. Hypothesizing the existence of undiscovered planets based on perceived irregularities rather than mathematical certainty.
This is how science works. You have data, you try and explain it by making guesses and hypothesis and then you TEST to find out if your hypothesis stands up or not. Then you have a theory. In the real world you don't always have perfect data and NOTHING is ever a mathematical certainty.
If we never guessed or imagined, never looked under rocks or in shadows then we would still be stumbling around in caves.
As for Einstein, not everyone agreed with that one either:
If there was any evidence Einstein was wrong it would be massive news. It's the most tested and proven theory we have, perhaps with the exception of Quantum Mechanics. Your references are all flawed.
Relativity and the Motion of Mercury
Charles Lane Poor, Ph.D.
Professor Emeritus of Celestial Mechanics,
Columbia University
Link to Paper (http://www.gsjournal.net/ScienceJournals/Historical%20PapersAstrophysics/Download/3394). From the Introduction:
“ Does the relativity theory, as asserted by Einstein, explain and account for even the single motion of tile perihelion of Mercury? In what way do the formulas of relativity differ from those of the classical mathematics of Newton, and how do these new formulas explain this motion? It is the purpose of this paper to discuss this single phase of the matter; to show that the very equations, or formulas, cited by the relativists as furnishing an explanation of this motion, utterly fail to furnish such an explanation. The formulas of relativity dynamics can not and do not explain the observed perihelial motion of Mercury. ”
This is from 1925. We have since proven Einstein right beyond a doubt with much better measurements and countless experiments.
The Theory of Mercury’s Anomalous Precession
Roger A. Rydin, Sc.D.
Associate Professor Emeritus of Nuclear Engineering,
University of Virginia
Link to Paper (http://www.tychos.info/citation/126A_MercuryPrecession.pdf)
Abstract: “ Urbain Le Verrier published a preliminary paper in 1841 on the Theory of Mercury, and a definitive paper in 1859. He discovered a small unexplained shift in the perihelion of Mercury of 39” per century. The results were corrected in 1895 by Simon Newcomb, who increased the anomalous shift by about 10%. Albert Einstein, at the end of his 1916 paper on General Relativity, gave a specific solution for the perihelion shift which exactly matched the discrepancy. Dating from the 1947 Clemence review paper, that explanation and precise value have remained to the present time, being completely accepted by theoretical physicists as absolutely true. Modern numerical fittings of planetary orbits called Ephemerides contain linearized General Relativity corrections that cannot be turned off to see if discrepancies between observation and computation still exist of the magnitude necessary to support the General Relativity estimates of the differences.
The highly technical 1859 Le Verrier paper was written in French. The partial translation given here throws light on Le Verrier’s analysis and thought processes, and points out that the masses he used for Earth and Mercury are quite different from present day values. A 1924 paper by a professor of Celestial Mechanics critiques both the Einstein and the Le Verrier analyses, and a 1993 paper gives a different and better fit to some of Le Verrier’s data. Nonetheless, the effect of errors in planet masses seems to give new condition equations that do not change the perihelion discrepancy by a large amount. The question now is whether or not the excess shift of the perihelion of Mercury is real and has been properly explained in terms of General Relativity, or if there are other reasons for the observations. There are significant arguments that General Relativity has not been proven experimentally, and that it contains mathematical errors that invalidate its predictions. Vankov has analyzed Einstein’s 1915 derivation and concludes that when an inconsistency is corrected, there is no perihelion shift at all! ”
This paper has a grand total of two citations, and none of those papers are refereed to by anyone else. This is pretty much a dead paper, and if it proved Einstein wrong or even threw serious doubt then it would be referenced and discussed all over. Poking holes in Relativity is one of science's Holy Grails after all and this would be front page science news if it could show just the hint of a crack.
Einstein’s Explanation of Perihelion Motion of Mercury
Hua Di
Academician, Russian Academy of Cosmonautics
Research Fellow (ret.), Stanford University
From Unsolved Problems in Special and General Relativity (https://web.archive.org/web/20190626004144/https://pdfs.semanticscholar.org/1d79/77c98a75068a913707aaec41b31ad967d562.pdf)
p.7
“ Einstein’s general theory of relativity cannot explain Mercury’s perihelion motion. He obtained “for the planet Mercury, a perihelion advance of 43” per century” by an incorrect integral calculus and many arbitrary approximations. His formula (1) is a poorly patched wrong result, tailored specially for Mercury. That is why his formula (1) fails to explain the perihelion motions for Earth and Mars. Einstein was unfair to blame “the small eccentricities of the orbits of these planets” for his failure. To sum up, Einstein’s general theory of relativity is dubious. ”
This isn't a paper, it's a book. The quote you are referencing is from something called "Einstein’s Explanation of Perihelion Motion of Mercury" by Hua Di which seems to only exist in this book, as I can't find it published anywhere and every reference leads back to this book.
Not a proper reference.

Also, the claimed Discovery of Neptune success of Perturbation Theory was discredited by some, who declared that they had only discovered Neptune by luck.
From what I understand there was an element of luck in that Neptune happened to be in a point in its orbit where it was affecting the orbit of Uranus in such a way that they predicted something was "out there". The something ended up being Neptune.
The 'n' body problem is solved by splitting it into a series of 2 body problems. Ultimately the test of the model is whether it yields accurate results. And it demonstrably does. We were able to predict the path of the solar eclipse in 2017 to the block level. We have sent craft to Mars and Pluto and landed one on a meteor because our models of the solar system are good enough that we can do that.
Your issue seems to be that our model isn't perfect, which is true. But it's demonstrably good enough to be useful.

This is from 1925.
Einstein's theory is older than that. What's your point? As far as I can see Charles Lane Poor was a Professor of Celestial Mechanics and Einstein was not, which is a more direct authority on Mercury than a theoretical physicist.
This paper has a grand total of two citations, and none of those papers are refereed to by anyone else.
Actually I see that this modern paper cites Charles Lane Poor's work, who you rejected for being 'too old'. Also, much of Einstein's work went uncited (https://forum.tfes.org/index.php?topic=14745.msg192478#msg192478). That's not a gauge for the validity or invalidity of a work.
This isn't a paper, it's a book.
It's a collection of papers published in a book. That seems like a weak argument.
We have since proven Einstein right beyond a doubt with much better measurements and countless experiments.
That's not true at all. Einstein was highly disputed, which is why he did not win the Nobel Prize for relativity, and was only awarded one for his work on the photoelectric effect, to which he responded by claiming racism.
...
Also, I found the current model of the Moon in the paper that stack posted (http://sites.apam.columbia.edu/courses/ap1601y/MoonEarthSin%20RMP.70.589.pdf).
See the illustration on page 600 and the caption, that the basic model was "adopted ever since."
V. THE MANY MOTIONS OF THE MOON
A. The traditional model of the Moon
A plane through the center of the Earth is determined at an inclination g of about 5 degrees with respect to the ecliptic. The Moon moves around the Earth in that plane on an ellipse with fixed semimajor axis a and eccentricity « of about 1/18. The Greek model was quite similar, except that the ellipse was replaced by an eccentric circle.
The plane itself rotates once every 18 years in the backward direction, i.e., against the prevailing motion in the solar system, while keeping its inclination constant. The perigee of the Moon, its point of closest approach to the Earth, makes a complete turn in the forward direction in about nine years.
The following picture (see Fig. 1) emerges: first we fix the direction of the spring equinox or some fixed star near it as the universal reference Q in the ecliptic: counting always from west to east, we determine the angle h from Q to the ascending node, i.e., the line of intersection for the Moon’s orbit with the ecliptic where the Moon enters the upper side of the ecliptic; from there we move by an angle g in the Moon’s orbital plane until we meet the perigee of the Moon; and finally we get to the Moon by moving through the true anomaly f. All these three angles have a double time dependence: linear (increasing for f and g, while decreasing for h) plus various periodic terms that average to 0.
(https://i.imgur.com/HpCH7Pn.png)
~
D. The evection—Greek science versus Babylonian astrology
The Babylonians knew that the full moons could be as much as 10 hours early or 10 hours late; this is due to the eccentricity of the Moon’s orbit. But the Greeks wanted to know whether the Moon displays the same kind of speedups and delays in the half moons, either waxing or waning. The answer is found with the help of a simple instrument that measures the angle between the Moon and the Sun as seen from the Earth. The half moons can be as much as 15 hours early or late. With the Moon moving at an average speed of slightly more than 308 per hour (its own apparent diameter!), it may be as much as 5° ahead or behind in the new/full moons; but in the half moons, it may be as much as 7°308 ahead or behind its average motion. This new feature is known as evection.
Ptolemy found a mechanical analog for this peculiar complication, called the crank model. It describes the angular coupling between Sun and Moon correctly, but it has the absurd consequence of causing the distance of the Moon from the Earth to vary by almost a factor of 2. In the thirteenth century Hulagu Khan, a grandson of Genghis Khan, asked his vizier, the Persian allround genius Nasir eddin al Tusi, to build a magnificent observatory in Meragha, Persia, and write up what was known in astronomy at that time. Ptolemy’s explanation of the evection was revised in the process. In the fourteenth century Levi ben Gerson of Avignon in southern France seems to have been the first astronomer to measure the apparent diameter of the Moon (see Goldstein, 1972, 1997). Shortly thereafter Ibn alShatir of Damascus in Syria proposed a model for the Moon’s motion that coincides with the theory of Copernicus two centuries later. The crank model was replaced by two additional epicycles, yielding a more elaborate Fourier expansion in our modern terminology (see Swerdlow and Neugebauer, 1984).
With the improvements of the Persian, Jewish, and Arab astronomers, as well as Copernicus, the changes in the Moon’s apparent diameter are still too large with +/ 10%. As in Kepler’s second law, the Fourier expansion (12) has to include epicycles both in the backward and in the forward direction, in the ratio 3:1.
"The crank model was replaced by two additional epicycles, yielding a more elaborate Fourier expansion in our modern terminology."
"With the improvements of the Persian, Jewish, and Arab astronomers, as well as Copernicus, the changes in the Moon’s apparent diameter are still too large with +/ 10%. As in Kepler’s second law, the Fourier expansion (12) has to include epicycles both in the backward and in the forward direction, in the ratio 3:1."

We have since proven Einstein right beyond a doubt with much better measurements and countless experiments.
That's not true at all. Einstein was highly disputed, which is why he did not win the Nobel Prize for relativity, and was only awarded one for his work on the photoelectric effect, to which he responded by claiming racism.
JSS wrote "We have since proven Einstein right beyond a doubt with much better measurements and countless experiments."
But all you say is "Einstein was highly disputed" which does not refute the statement made by LSS in the slightest.
It is true that originally "Einstein was highly disputed" but since 1915 Einstein's General Theory of Relativity has gained almost, but not quite, universal acceptance.
There is now voluminous experimental evidence supporting GRT. for example:
Experimental Tests of General Relativity by Slava G. Turyshev (https://arxiv.org/pdf/0806.1731.pdf)
Why Relativity's True: The Evidence for Einstein's Theory[/b]]Why Relativity's True: The Evidence for Einstein's Theory (http://[b)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Never stop testing
Even with all that evidence, we continue to put general relativity to the test. Any sign of a crack in Einstein's magnificent work would spark the development of a new theory of gravity, perhaps paving the way to uncovering the full quantum nature of that force. That's something we currently don't understand at all.
But in all regards, GR passes with flying colors; from sensitive satellites (https://www.space.com/456einsteinwarpedviewspaceconfirmed.html) to gravitational lensing (https://www.space.com/40958einsteingeneralrelativitytestdistantgalaxy.html), from the [urlhttps://www.space.com/37745einsteinrelativitytestedbystarblackhole.html]orbits of stars[/url] around giant black holes to ripples of gravitational waves (https://www.space.com/31922gravitationalwavesdetectionwhatitmeans.html) and the evolution of the universe itself (https://www.space.com/13347bigbangoriginsuniversebirth.html), Einstein's legacy is likely to persist for quite some time.
Not that anyone regards Einstein's theories as the be all and end all.
Also, I found the current model of the Moon in the paper that stack posted (http://sites.apam.columbia.edu/courses/ap1601y/MoonEarthSin%20RMP.70.589.pdf).
See the illustration on page 600 and the caption, that the basic model was "adopted ever since."
That illustration might be but the current lunar orbit is no longer described in terms of an epicycle and different (with possibly an equant) but as an approximate precessing ellipse.
V. THE MANY MOTIONS OF THE MOON
A. The traditional model of the Moon
A plane through the center of the Earth is determined at an inclination g of about 5 degrees with respect to the ecliptic. The Moon moves around the Earth in that plane on an ellipse with fixed semimajor axis a and eccentricity « of about 1/18. The Greek model was quite similar, except that the ellipse was replaced by an eccentric circle.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D. The evection—Greek science versus Babylonian astrology
The Babylonians knew that the full moons could be as much as 10 hours early or 10 hours late; this is due to the eccentricity of the Moon’s orbit. But the Greeks wanted to know whether the Moon displays the same kind of speedups and delays in the half moons, either waxing or waning. The answer is found with the help of a simple instrument that measures the angle between the Moon and the Sun as seen from the Earth. The half moons can be as much as 15 hours early or late. With the Moon moving at an average speed of slightly more than 308 per hour (its own apparent diameter!), it may be as much as 5° ahead or behind in the new/full moons; but in the half moons, it may be as much as 7°308 ahead or behind its average motion. This new feature is known as election.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
"With the improvements of the Persian, Jewish, and Arab astronomers, as well as Copernicus, the changes in the Moon’s apparent diameter are still too large with +/ 10%. As in Kepler’s second law, the Fourier expansion (12) has to include epicycles both in the backward and in the forward direction, in the ratio 3:1."
But all you include in your quote is "V. The traditional model of the Moon" and ignore the rest:
"VI. Newton’s Work in Lunar Theory",
"VII. Lunar Theory in the Age of Enlightenment",
"VIII. The Systematic Development of Lunar Theory",
"IX. The Canonical Formalism",
"X. Expansion around a Periodic Orbit" and most importantly
"XI. Lunar Theory in the 20th Century".
And soon after Newton published his "Laws of Motion and Universal Gravitation" Newton and the astronomers of his day put in a great deal of effort into explaining the details of the lunar orbit.
They had allowed for the quadripole gravitational moment of Earth due to its oblateness and the effect of the Sun's gravitation but it didn't match until the inc;used the effect of Jupiter's gravitation.
In the end it was a great triumph of Newton's Laws and finally, most astronomers accepted their accuracy.
It's so interesting that from "The traditional model of the Moon" even the ancient Greeks and Ptolemy were very close to the modern orbital characteristics of the Moon but seems nothing like the orbit or the moon in tour flat Earth model. Why is that?
So none of what you quote represents "Lunar Theory in the 20th Century" other than the diagram which shows the inclination of the lunar orbit at 5° very close to the modern 5.15°.
(https://upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Lunar_Orbit_and_Orientation_with_respect_to_the_Ecliptic.tif/lossypage11920pxLunar_Orbit_and_Orientation_with_respect_to_the_Ecliptic.tif.jpg)
Why are you so far out of date with your references?
A great deal has been learned since the time of Isaac Newton and even since Einstein first published his General Theory of Relativity.

JSS wrote "We have since proven Einstein right beyond a doubt with much better measurements and countless experiments."
But all you say is "Einstein was highly disputed" which does not refute the statement made by LSS in the slightest.
It is true that originally "Einstein was highly disputed" but since 1915 Einstein's General Theory of Relativity has gained almost, but not quite, universal acceptance.
There is now voluminous experimental evidence supporting GRT. for example:
Acceptance != proven. There were points in history where people accepted the existence of witches too. The three proofs Einstein gave for GR were pretty disputed, and still are criticized by physicists if you care to look.
That illustration might be but the current lunar orbit is no longer described in terms of an epicycle and different (with possibly an equant) but as an approximate precessing ellipse.
Epicycles are still used under the name of Fourier series, which is discussed in the previous quote I gave.
But all you include in your quote is "V. The traditional model of the Moon" and ignore the rest:
"VI. Newton’s Work in Lunar Theory",
"VII. Lunar Theory in the Age of Enlightenment",
"VIII. The Systematic Development of Lunar Theory",
"IX. The Canonical Formalism",
"X. Expansion around a Periodic Orbit" and most importantly
"XI. Lunar Theory in the 20th Century".
And soon after Newton published his "Laws of Motion and Universal Gravitation" Newton and the astronomers of his day put in a great deal of effort into explaining the details of the lunar orbit.
They had allowed for the quadripole gravitational moment of Earth due to its oblateness and the effect of the Sun's gravitation but it didn't match until the inc;used the effect of Jupiter's gravitation.
In the end it was a great triumph of Newton's Laws and finally, most astronomers accepted their accuracy.
As mentioned earlier, the end of that paper says that Newton's laws are not a sufficient explanation:
The threebody problem teaches us a sobering lesson about our ability to comprehend the outside world in terms of a few basic mathematical relations. Many physicists, maybe early in their careers, had hopes of coordinating their field of interest, if not all of physics, into some overall rational scheme. The more complicated situations could then be reduced to some simpler models in which all phenomena would find their explanation. This ideal goal of the scientific enterprise has been promoted by many distinguished scientists [see Weinberg’s (1992) Dream of a Final Theory, with a chapter ‘‘Two Cheers for Reductionism’’]
~
Many physicists may be tempted to see in Newton’s equations of motion and his universal gravitation a sufficient explanation for the threebody problem, with the details to be worked out by the technicians. But even a close look at the differential equations (29) and (30) does not prepare us for the idiosyncracies of the lunar motion, nor does it help us to understand the orbits of asteroids in the combined gravitational field of the Sun and Jupiter.
The "threebody problem teaches us a sobering lesson"... "Many physicists may be tempted to see in Newton’s equations of motion and his universal gravitation a sufficient explanation"... An admission that it can't really explain the situation.

The "threebody problem teaches us a sobering lesson"... "Many physicists may be tempted to see in Newton’s equations of motion and his universal gravitation a sufficient explanation"... An admission that it can't really explain the situation.
Absolutely not. It's the same mistake you make over and over, but I admire your perseverance. That's the "perfect solution fallacy": because there are limitations and difficulties, you seem to assume it doesn't work at all. It would be like saying that because we'll never know the exact numerical value of pi, we really don't have any idea of its value and it could as well be 2 or 5. Just because we don't know everything doesn't mean we don't know anything.
No serious physicist ever said Nbody simulators can't be made, because they do exist and they do work very well.

The "threebody problem teaches us a sobering lesson"... "Many physicists may be tempted to see in Newton’s equations of motion and his universal gravitation a sufficient explanation"... An admission that it can't really explain the situation.
Absolutely not. It's the same mistake you make over and over, but I admire your perseverance. That's the "perfect solution fallacy": because there are limitations and difficulties, you seem to assume it doesn't work at all. It would be like saying that because we'll never know the exact numerical value of pi, we really don't have any idea of its value and it could as well be 2 or 5. Just because we don't know everything doesn't mean we don't know anything.
The author doesn't say it comes close either. He gave up on it and says that physicists may be tempted to see that Newton's laws provide a sufficient explanation, but that is not the case. He says that Newton's laws do not provide a sufficient explanation.
You are only speculating that the attempts come close, claiming that "he got close enough" or whatever, when this is not what is said at all.
No serious physicist ever said Nbody simulators can't be made, because they do exist and they do work very well.
They say that the Nbody orbits require symmetrical systems. Here is physicist David Gozzard:
https://www.youtube.com/watch?v=eqSPvyaxMI8
“ Three hundred and fifty years ago Isaac Newton formulated his theory of gravity Newton's theory unified the heavens and the earth under the same physical laws and neatly explained the orbits of the planets, the motion of comets, and how the moon causes the tides. Although Einstein's general relativity has supplanted Newton's theory as a better model of gravity, Newtonian mechanics got us to the moon and are still used to calculate the trajectories of spacecraft throughout the solar system. In spite of this success and more than three centuries of progress we still do not have a neat set of equations that allow us to calculate the orbital parameters of more than two objects at once. This is called the threebody problem, or more generally the nbody problem. ”
"this is a problem because there are certainly more than two bodies in the solar system. We can get around this by formulating equations that give an approximate solution for motions of three or more objects and since the mid twentieth century we've been able to use computers to simulate the orbits using brute force step by step calculations"
So no, they admit that it's not possible and that they need to use approximations and 'brute force' the orbits.
Author goes on to discuss the discovered orbits:
(https://i.imgur.com/docrXq0.png)
"any slight perturbation to one of the objects will result in chaos with the body either crashing into its orbital partners or being ejected from the group entirely"
And when going over further discovered symmetrical orbits:
"While physically possible, each of these orbits is more unlikely to exist than the last"
That's the state of Newtonian astronomy.
There are some special things they can do to make it look like a Moon going around a planet around a sun, but those don't work out either. They use the Restricted Three Body Problem, where one of the bodies is of zero mass. Even then, the 'Moon' is still chaotic. The benefit of the Restricted Three Body Problem and the Massless moon meant that that the moon would be no longer ejected from the system, as it would usually be. It is confined to what is known as "Hill's Region".
From Scholarpedia: http://www.scholarpedia.org/article/Three_body_problem by Dr. Alain Chenciner
(https://wiki.tfes.org/images/thumb/8/83/Hill%27s_Region.gif/500pxHill%27s_Region.gif)
The above depicts a crazy and chaotic moon which even makes a uturn in mid orbit.
From the text that accompanies the image:
“ 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.
Even in this weird version with a massless Moon, it doesn't really work.
From Computing the long term evolution of the solar system with geometric numerical integrators (https://publications.mfo.de/handle/mfo/1355) by Shaula Fiorelli Vilmart and Gilles Vilmart we see a three body simulation of the SunEarthMoon system created by mathematicians which does use the actual supposed masses of the Earth, Moon and Sun.
Abstract: “ Simulating the dynamics of the Sun–Earth–Moon system with a standard algorithm yields a dramatically wrong solution, predicting that the Moon is ejected from its orbit. In contrast, a well chosen algorithm with the same initial data yields the correct behavior. We explain the main ideas of how the evolution of the solar system can be computed over long times by taking advantage of socalled geometric numerical methods. Short sample codes are provided for the Sun–Earth–Moon system. ”
Sure enough, with the standard algorithm, it fell apart:
(https://wiki.tfes.org/images/thumb/3/3b/SunEarthMoon.gif/400pxSunEarthMoon.gif)
The paper explains that sympectic integrators are necessary and shows a simulation with a working Moon along side this one.
Symplectic integrators are used in particle physics as well (which also has trouble with stability). This abstract of a particle physics paper says (https://accelconf.web.cern.ch/IPAC2015/papers/mopma030.pdf):
“ It has been long understood that long time single particle tracking requires symplectic integrators to keep the simulations stable ”
So again, we see that cheats are necessary.

Well, it's still the same fallacy. You take a video of a guy who says it DOES work with computer simulations even though we don't have what he calls "a neat set of equations" (ie, a formal closedform solution), he literally says "we can get around this". From that video, you infer that he admits it does not work. If you did that on Wikipedia, you would eventually get banned for backing your statements with citations that say exactly the opposite.
"It's an approximation" still isn't an equivalent of "it doesn't work". Perfect solution fallacy, once again.

Well, it's still the same fallacy. You take a video of a guy who says it DOES work with computer simulations even though we don't have what he calls "a neat set of equations" (ie, a formal closedform solution), he literally says "we can get around this". From that video, you infer that he admits it does not work. If you did that on Wikipedia, you would eventually get banned for backing your statements with citations that say exactly the opposite.
"It's an approximation" still isn't an equivalent of "it doesn't work". Perfect solution fallacy, once again.
"We can get around this" means that he needs to employ work arounds and cheats. Two body approximations for a three body system is hardly a solution to the three body problem, for example. By treating it as two problems, Earth and Moon and Sun and Earth, it totally ignores the problem of three bodies.
Appealing to a word "approximation" alone is not reasonable argument.
You need to show us what they are doing and demonstrate that the three body problem actually does work, to some degree, to explain the motions of the celestial systems. Rather, you are appealing to the word "approximations" without knowledge of what is actually being "approximated".
The number "one" is approximately "zero," but this is also rather dumb to state as a rule and invalidates a proof which shows that 1 + 1 = 2.
You are arguing that we can cheat our way through. Lets see the nature of these cheats you find acceptable.
But you guys cannot actually show us a working system, or a statement from a physicist which states what you are arguing, and must argue on basis of personal "inferences" of the statements that you read. Rather insufficient, TBH.

I really don't get your point. An approximated solution is an approximated solution. It's not an absence of solution. Just like 3.14159265 is an approximated value of pi. It will be accurate enough for most purposes. If for some special purpose I need a more accurate value, I'll take the time and effort needed to find one.
We have approximated solutions to the nbody problem and they are accurate enough for the systems we want to simulate. It looks like you really don't want that to be true, but facts are stubborn things.

From the askamathematician page we saw that it was acknowledged that the three body problem doesn't work except for some goofy scenerios, and then the author went off on tangents talking about cheating ways of doing it:
https://www.askamathematician.com/2011/10/qwhatisthethreebodyproblem/
the problem with the 3body problem is that it can’t be done, except in a very small set of frankly goofy scenarios (like identical planets following identical orbits).
...
So, if you want to calculate the orbits of all the planets, a “2body approximation” will get you more than 99% of the way to the right answer.
...
But even with just mechanical pencil and paper there are cheats. For example, although there are more than three bodies in the solar system (the Sun, eight planets, dozens of moons, and millions of asteroids and comets), almost everything behaves, roughly, as though it were in a two body system.
Two body approximations, cheats, etc.
How is a two body approximation a solution to this? This ignores the problem of three bodies and is a total cheat.
Again, lets see the nature of these "approximation" cheats that you claim exist, and are putting forward.
How about some real sources for what you are claiming for once, rather than citing yourself as a source?

From the askamathematician page we saw that it was acknowledged that the three body problem doesn't work except for some goofy scenerios, and then the author went off on tangents talking about cheating ways of doing it:
https://www.askamathematician.com/2011/10/qwhatisthethreebodyproblem/
the problem with the 3body problem is that it can’t be done, except in a very small set of frankly goofy scenarios (like identical planets following identical orbits).
...
So, if you want to calculate the orbits of all the planets, a “2body approximation” will get you more than 99% of the way to the right answer.
...
But even with just mechanical pencil and paper there are cheats. For example, although there are more than three bodies in the solar system (the Sun, eight planets, dozens of moons, and millions of asteroids and comets), almost everything behaves, roughly, as though it were in a two body system.
Two body approximations, cheats, etc.
How is a two body approximation a solution to this? This ignores the problem of three bodies and is a total cheat.
Again, lets see the nature of these "approximation" cheats that you claim exist, and are putting forward.
How about some real sources for what you are claiming for once, rather than citing yourself as a source?
OK, I'll begin with the same askamathematician page you quoted:
Despite that, we do alright, and happily, reality doesn’t concern itself with doing math, it just kinda “does”. For example, quantum field theory, despite being the most accurate theory that ever there was, never involves exactly solving anything. Once a physicist gets a hold of all the appropriate equations and a big computer, they can start approximating things. With enough computing power and time, these approximations can be made amazingly good. Computer simulation and approximation is a whole science unto itself.
Point is, this effect only shows up in systems with three or more bodies, it’s chaotic (in the chaos theory sense), and there is no way to predict it exactly. That being said, we can still get computers to come pretty close (up to a point, because chaos is a punk), and there are even some mathematical tricks to get reasonable solutions that, while not perfect, are still pretty good (and can even get us well into that last “1% of weirdness”).

It's saying that the approximations can get pretty good, and only gives the method of the two body problem approximation for the orbital problems in the article.
This invalidates the inferences that you had to argue from.
Again, you guys just read what you want to read. Those quotes are in the section talking about approximations. The approximations can be good, but ultimately invalid and irrelevant for the three body problem.

The source you chose literally says that the lack of a perfect formal solution isn't a big problem because we can get computer simulations to be very accurate. If you decide to read and understand anything else from this text, I'm afraid there isn't anything more I can say.
Edited for your edit: why would an approximation be irrelevant for the nbody problem, or any other problem? You never get a perfect solution from a computer simulation. The perfect solution is the real system.

The source you chose literally says that the lack of a perfect formal solution isn't a big problem because we can get computer simulations to be very accurate. If you decide to read and understand anything else from this text, I'm afraid there isn't anything more I can say.
Edited for your edit: why would an approximation be irrelevant for the nbody problem, or any other problem? You never get a perfect solution from a computer simulation. The perfect solution is the real system.
The only orbital approximation method discussed in the article is talking about two body problem approximations. They can approximate it by treating it as SunEarth and EarthMoon.
The approximation is basically "The number two is close to the number three, so oh well, approximated"
It is bewildering how you can defend this as "not a perfect solution" but that it's almost there, and quote a section which calls those approximations "pretty good" and think that this is any way sufficient. The approximation method totally ignored the concept of three bodies.

The only orbital approximation method discussed in the article is talking about two body problem approximations.
This is absolutely not what the article says. Read again:
Point is, this effect only shows up in systems with three or more bodies, it’s chaotic (in the chaos theory sense), and there is no way to predict it exactly. That being said, we can still get computers to come pretty close (up to a point, because chaos is a punk), and there are even some mathematical tricks to get reasonable solutions that, while not perfect, are still pretty good (and can even get us well into that last “1% of weirdness”).
This "last 1% of weirdness" being what you can't explain with a 2 body simulation.
I rest my case, perfect solution fallacy. Anyway, a simulation is pretty much always an approximation. The map is not the territory. Ceci n'est pas une pipe.

No, it's already talking about approximations by that point in the article.
The previous paragraph even abandons the idea of any bodies:
"When you have even more bodies you can almost abandon the idea that there are any bodies at all, and move over to fluid dynamics. Although, again, that’s just an approximation."
And before that " Once a physicist gets a hold of all the appropriate equations and a big computer, they can start approximating things. With enough computing power and time, these approximations can be made amazingly good. Computer simulation and approximation is a whole science unto itself."
By the time of your quote it's talking about how we can approximate our way out of it
The only approximation mentioned in the article is two body approximations. And how computer simulations can be approximated.
So, since your quote comes from a section about approximations, how does this help you?

No, it's already talking about approximations by that point in the article.
The whole point of this article is that approximations are good enough for our use cases. Including for the nbody problem or other problems in physics that may not have a "perfect" mathematical solution either. You distort it to keep on with the same perfect solution fallacy and claim that since there is no perfect solution, there is no solution at all. This continues to be wrong, no matter how many times you repeat it.

[You] claim that since there is no perfect solution, there is no solution at all.
Ah, how refreshing to hear that from an RE'er. Now, if only you could convince the rest of your side of this...
That said, patternbased approximations are largely modelagnostic. If this is the best you have, then you no longer have an argument for RE's predictive powers being superior in this case.
This continues to be wrong, no matter how many times you repeat it.
Hey! That's my line! >:(

[You] claim that since there is no perfect solution, there is no solution at all.
Ah, how refreshing to hear that from an RE'er. Now, if only you could convince the rest of your side of this...
That said, patternbased approximations are largely modelagnostic. If this is the best you have, then you no longer have an argument for RE's predictive powers being superior in this case.
Depends on how approximate the predictions can get, though, right? We can regularly predict with exceptional accuracy everything that happens in our solar system among the celestial bodies we know of  eclipses, comets, etc.  but more spectacularly the arrival of dozens of manmade objects sent off to various places such as a single moon of a distant planet. That would be quite a trick if NASA and others weren't able to actually predict these things, but were just amazing guessers.

That would be quite a trick if NASA and others weren't able to actually predict these things, but were just amazing guessers.
You've missed the point. The question isn't whether those predictions are accurate (even if imprecise), but what the source of them is. RE'ers like to claim that it's RET, but in this thread we have the smoking gun  it's an observation of patterns and computer modelling based on those patterns. Accuracy aside, as soon as you make this admission, it ceases to be evidence pointing towards RET.

That would be quite a trick if NASA and others weren't able to actually predict these things, but were just amazing guessers.
You've missed the point. The question isn't whether those predictions are accurate (even if imprecise), but what the source of them is. RE'ers like to claim that it's RET, but in this thread we have the smoking gun  it's an observation of patterns and computer modelling based on those patterns. Accuracy aside, as soon as you make this admission, it ceases to be evidence pointing towards RET.
But NASA models not using patterns but 'RET' models (Newtonian Gravity, Relativity etc.). They take the equations of motion from these theories and put them into some code  am I missing something?

That would be quite a trick if NASA and others weren't able to actually predict these things, but were just amazing guessers.
You've missed the point. The question isn't whether those predictions are accurate (even if imprecise), but what the source of them is. RE'ers like to claim that it's RET, but in this thread we have the smoking gun  it's an observation of patterns and computer modelling based on those patterns. Accuracy aside, as soon as you make this admission, it ceases to be evidence pointing towards RET.
But NASA models not using patterns but 'RET' models (Newtonian Gravity, Relativity etc.). They take the equations of motion from these theories and put them into some code  am I missing something?
That is my understanding as well. The "computer modelling" is not simple pattern recognition like some fancy AI, but simply highly complex and/or iterative math equations or something.

I think the actual interesting question in this discussion is are there any real, physical systems that are analytically solvable? I think the answer is no. Can we discount all of physics, now?

Acceptance != proven. There were points in history where people accepted the existence of witches too.
So firstly, no scientific theory is ever proven in the strictest sense. The best we can say is that Einstein's theories work better than Newton's in certain situations. And by "work better" I mean they make predictions which more accurately match observations. For most normal situations there is no measurable difference.
You get hung up on the 3 body problem a lot and I don't understand what your issue is.
You're right in that there is no solution. There are no equations which have a variable "t" for time, you plug in the initial positions of the bodies, you put in a value for t and the equations give you the results in terms of where all the bodies will be 't' seconds into the future. I think if anything this is a failure of mathematics, or maybe it's an inherent problem with chaotic systems (chaotic in the mathematical sense). The weather is chaotic too  or our models of it are. That's why weather forecasts are often wrong and there is no such things as a truly long range forecast.
The best we can do is split the nbody problem into a series of 2 body problems. Solve those for a small time increment and then iterate. So yes, it's a bit of a workaround but the key here is it does work. It's got us to the moon, it's put craft on Mars. I don't think any of these models are simply using cycles, they're using Newton's equations to work out from initial positions and velocities what the next positions and velocities will be in small time increments and iterating. But because of this iteration if the initial values are incorrect  in the sense that they are not perfect, which they can't be  then over time those errors will build up.
So yes, this is a weakness of our models, BUT...a model does not have to be perfect to be useful. Someone else said that the value of pi cannot be used perfectly because it's an irrational number. But we know the value well enough to use it to make accurate calculations. If we used the value 3 then that wouldn't be good enough for most purposes, 3.14159265 (which is as much as I know off the top of my head) is good enough for pretty much any purpose.
The headline is while yes, we don't have "a solution" to the n body problem I don't see why you think that is a problem. We have ways of calculating it well enough to send craft into space and land them on other celestial bodies. I'd say that's a triumph of our models, not a failure.

I think the actual interesting question in this discussion is are there any real, physical systems that are analytically solvable? I think the answer is no. Can we discount all of physics, now?
I am not sure I fully understand what it means for something to be "analytically solvable." But if what you are writing here is true, would your statement then equally apply to proposed FE physical systems?

am I missing something?
You are. Namely: the source of these "approximate" equations.

am I missing something?
You are. Namely: the source of these "approximate" equations.
The equations aren't approximations  just their solutions.