[Tom Bishop]

Given the excellent quality of the rebuttals

I don't see any quality. AATW, Tunemi, SteelyBob are citing themselves as their source versus the physicists who say directly that the three body problem does not work.

Professor Ashish Tewari said "we cannot mathematically prove certain observed facts (such as the stability of the solar system) concerning N-body motion"

They simply can't do it. Don't pretend that you are a better authority than he is.

[WTF_Seriously]

Professor Ashish Tewari said "we cannot mathematically prove certain observed facts (such as the stability of the solar system) concerning N-body motion"

Did you miss the little point I emphasized?  Funny how he didn't go on to say that this definitively proves the n-body system is false.  I wonder why.

You can't even give the equation of a single curved line.

[stevecanuck]

Given the excellent quality of the rebuttals

I don't see any quality. AATW, Tunemi, SteelyBob are citing themselves as their source versus the physicists who say directly that the three body problem does not work.

Professor Ashish Tewari said "we cannot mathematically prove certain observed facts (such as the stability of the solar system) concerning N-body motion"

They simply can't do it. Don't pretend that you are a better authority than he is.

They seemed to explain that quite thoroughly. I guess pi doesn't exist because an exact number can't be determined for it.

Anyway, were still waiting to hear of the other "many problems".

[Tumeni]

I don't see any quality. AATW, Tunemi, SteelyBob are citing themselves as their source versus the physicists who say directly that the three body problem does not work.

No, I did not. I referred to activities which showed that it works well enough to send stuff around and to other planets. The citations are in libraries worldwide, and in multiple records, in multiple form, of multiple space flights.

AATW said much the same, as far as I'm concerned.

[SteelyBob]

Given the excellent quality of the rebuttals

I don't see any quality. AATW, Tunemi, SteelyBob are citing themselves as their source versus the physicists who say directly that the three body problem does not work.

Professor Ashish Tewari said "we cannot mathematically prove certain observed facts (such as the stability of the solar system) concerning N-body motion"

They simply can't do it. Don't pretend that you are a better authority than he is.

Are you seriously, publicly doubling down on this one? Unbelievable. Nice appeal to authority at the end there as well, which is of course both a schoolboy debating fallacy and completely misses the mark, given that we all entirely agree with the authority in question, whereas you just don't understand or are wilfully misrepresenting what he is saying.

It's pretty astonishing that as a flat earth proponent you are quoting somebody who is referring to the stability of the solar system as one of many 'certain observed facts', but let's park that for the time being.

Tewari's quote in full, from your own wiki:

In the next section, it will be shown that two additional integrals can be obtained when N = 2 from the considerations of relative motion of the two bodies. Hence, a two-body problem is analytically solvable. However, with N > 2, the number of unknown motion variables exceeds the total number of integrals; thus, no analytical solution exists for the N-body problem when N > 2. Due to this reason, we cannot mathematically prove certain observed facts (such as the stability of the solar system) concerning N-body motion. The best we can do is to approximate the solution to the N-body problem either by a set of two-body solutions or by numerical solutions.

The problem, as he himself explains in your own wiki quote , is not that planets can't do what they are observed to do, it's that you simply can't solve the problem algebraically. As I pointed out, that is true of countless complex science and engineering problems involving PDEs, and is in no way indicative of them not being true. If you really want citations, check out the Clay Millennium prizes, one of which is a particular aspect of the N-S equation solution. 21 years on, and nobody has claimed it. And yet our aircraft still fly...

[AATW]

I don't see any quality. AATW, Tunemi, SteelyBob are citing themselves as their source versus the physicists who say directly that the three body problem does not work.

Weirdly, you seem to be disagreeing with me while quoting someone who says exactly what I’m trying to explain. There is no analytical solution, but there are numerical solutions.
If you continue not to be able to understand this then I’m not sure how to help, this has been explained to you in multiple previous threads.

We clearly have very different definitions of something which “does not work”. The rover currently sitting on Mars and the pictures we got back from a probe sent to Pluto tells me that the solutions we do have work rather well. Meanwhile, I note you dodged the question about what predictive power FE has.

The authority you quote by the way is a Professor in a Department of Aerospace Engineering who has written multiple books about space flight, so do you cede to his authority on that and therefore the shape of the earth or are you simply cherry picking again?

[jack44556677]

Sure, ignore the three body paradox.

How about the creation of spherical balls of matter in the first place (in order to have any systems, of any number, at all).

None of the models can do that without invoking massive amounts of speculative fiction.

Stars, planets, you name it.  Not without "supermassive black holes" and worse...

[Tumeni]

Sure, ignore the three body paradox.

How about the creation of spherical balls of matter in the first place (in order to have any systems, of any number, at all).

None of the models can do that without invoking massive amounts of speculative fiction.

Stars, planets, you name it.  Not without "supermassive black holes" and worse...

We have the physical, observational, and anecdotal proof of its shape from over 60 years of space flights around our planet.

The presence of only speculation about its origin and formation is not a disproof of its current shape.

[jack44556677]

I don't accept data that requires such abject appeal to authority (cannot be validated / verified independently), but even assuming you do AND that the earth is a sphere (and everything else in the heavens, to boot) doesn't help us explain what is (or why it is) through theory.

I would like to be clear that I think this entire thread is a farce (very much related, if not quintessentially demonstrative, of what JSS is certain does not exist in the "zealous" thread). 

RET is not a thing and surely if it was it wouldn't contain the totality of all the universe within it.  We are merely criticizing creation mythology that is disingenuously/erroneously passed off as science from childhood.

[Tumeni]

I don't accept data that requires such abject appeal to authority (cannot be validated / verified independently)

That's great. Show us some data that you do accept, then, and show the independent validation/verification of it. More than one of us have asked for details of the "research" you have done. Ideal time to tell us.

[SteelyBob]

Sure, ignore the three body paradox.

It's not a paradox. It's called the 'three body problem'. It's a maths problem that can't be solved algebraically and has solutions that tend to be chaotic in nature, meaning that small perturbations in starting conditions result in large changes in outcome. It's not paradoxical.

How about the creation of spherical balls of matter in the first place (in order to have any systems, of any number, at all).

None of the models can do that without invoking massive amounts of speculative fiction.

Stars, planets, you name it.  Not without "supermassive black holes" and worse...

Whilst there are indeed huge aspects of how things came to be that remain unexplained, things like spherical planets etc are at least consistent and coherent with our current thinking. How a flat earth came to be would be equally mysterious and unanswered and, given the total lack of coherence and enormous logical leaps required, far more speculative and fictitious.

[Action80]

I see that you conceded that numerical solutions do not fully simulate gravity. If the greatest mathematicians have not been able to get this gravity system to work, why should anyone believe that this system exists?

This is utterly preposterous I'm afraid.

Firstly, you have completely misunderstood the papers you are quoting. In this one, for example - https://publications.mfo.de/handle/mfo/1355 - the 'standard algorithm' used to calculate the example where the moon is ejected from the three-body system is an explicit Euler solution, which is in itself a numerical method - a simplification needed to deal with the complexity of the set of PDEs. That you are holding this up as some sort of 'correct' answer and implying that the authors have tinkered with some parameters until they got to the answer they wanted shows a total failure to comprehend the information you have used in the wiki.

In the example in the paper the entire point was that, over time, implicit and explicit Euler methods do not conserve the energy of a system - there is a small error at each step that aggregates over time. This makes them poor choices for the long-term simulation of an n-body problem. The symplectic method used does conserve energy, and is therefore better in this regards, even though it is far from perfect.

Partial differential equations are extremely difficult to solve algebraically except for certain very specific cases, but they are everywhere in scientific and engineering problems at the micro and macro levels. It is impossible, for example, to solve the Navier Stokes equations for turbulent / viscous, compressible flow over an aircraft, but this is the information needed to accurately calculate lift and drag. And yet our aircraft don't fall from the sky; this is because aerodynamicists make judicious use of wind tunnels and various numerical methods, often involving vast computing power, to approximate solutions. Their inability to solve the equations by algebra does not make aviation impossible.   

There's nothing wrong with not understanding this stuff - it's complex, and well into undergraduate maths / physics / engineering territory - but it might be worth getting your assertions checked over by somebody who does understand it before publishing a wiki article about it.

Next time I believe you could spare us the long winded sentences and simply state the following:

"There is no numerical solution to the n-body problem."

[SteelyBob]

Next time I believe you could spare us the long winded sentences and simply state the following:

"There is no numerical solution to the n-body problem."

Well, no I couldn't, because that's almost the precise opposite of what I said. There are almost no algebraic solutions to the n-body problem, aside from certain very specific unique cases, and almost every other solution is a numeric one of some kind.

Moreover, it is important to understand that the example Tom is using to highlight his point is in itself a numeric solution - he has completely failed to understand that, or is deliberately misrepresenting it.

[iamcpc]

Because of;

The hundreds of years of practical experience of it,
the measurements which correspond with it, and
the extensive space travel, since the 1960s or so, which has had flight paths based upon it.

Here is an example:





at the 2.52 minute mark which represents our solar system according to modern physics I notice that none of the marbles turned into moons and orbited other planets.

at the 4:40 mark he tries to demonstrate the three body system but it sure looks to me like both marbles are just orbiting the "sun" in that example. One marble does not do more than bump into the other once or twice. They never form what to me looks like a three body system which exists in our solar system with the sun, the earth, and the moon.

From a mathematical perspective we are unable to explain how our moon orbits our earth while the entire earth moon system also orbits the sun.

If we are unable to explain how this three body system works questioning how our 293857298375 body solar system works is not so illogical

[SteelyBob]

at the 2.52 minute mark which represents our solar system according to modern physics I notice that none of the marbles turned into moons and orbited other planets.

at the 4:40 mark he tries to demonstrate the three body system but it sure looks to me like both marbles are just orbiting the "sun" in that example. One marble does not do more than bump into the other once or twice. They never form what to me looks like a three body system which exists in our solar system with the sun, the earth, and the moon.

That's a ridiculous interpretation of an experiment that is clearly intended as an illustrative teaching aid, and not an accurate simulation of things. For a start, the marbles are all the same size, and none of them are heavy enough to distort the fabric enough to have the effect needed.

From a mathematical perspective we are unable to explain how our moon orbits our earth while the entire earth moon system also orbits the sun.

If by 'mathematical perspective' you mean 'write down an equation that enables us to plug in variables and output position and velocity and data' then no, we can't, but that doesn't mean we can't explain what's going on, or that the planets don't do what we expect them to do. Indeed, we have very accurate predictive models for planetary motion based on the n-body formulae - we just acknowledge that they aren't perfect over long time frames due to both the slight inaccuracies inherent in numerical methods and the sensitivity of the long-term results to small changes in starting conditions, which we can't measure perfectly. If you require that extremely high bar to be reached by every aspect of science, then we would also have no explanation for the behaviour of molecules, atoms and sub-atomic particles, or how aircraft fly, to pick just two examples of things that can't be algebraically solved and require numerical method solutions.

If we are unable to explain how this three body system works questioning how our 293857298375 body solar system works is not so illogical

Questioning it is entirely reasonable; assuming that an entirely baseless alternative model for the shape and disposition of the earth and its solar system, with no supporting evidence, no predictive power, and a whole range of enormous observable contradictions, is in fact a better choice due to some extremely minor limitations at the frontiers of human understanding is just plain daft.

[scomato]

Because of;

The hundreds of years of practical experience of it,
the measurements which correspond with it, and
the extensive space travel, since the 1960s or so, which has had flight paths based upon it.

Here is an example:


https://www.youtube.com /watch?v=MTY1Kje0yLg


at the 2.52 minute mark which represents our solar system according to modern physics I notice that none of the marbles turned into moons and orbited other planets.

at the 4:40 mark he tries to demonstrate the three body system but it sure looks to me like both marbles are just orbiting the "sun" in that example. One marble does not do more than bump into the other once or twice. They never form what to me looks like a three body system which exists in our solar system with the sun, the earth, and the moon.

From a mathematical perspective we are unable to explain how our moon orbits our earth while the entire earth moon system also orbits the sun.

If we are unable to explain how this three body system works questioning how our 293857298375 body solar system works is not so illogical

Can you be more specific about what is wrong with the mathematical solutions to orbit calculations? This is material that would be covered in a 4th year undergraduate Physics course, well within reach of understanding by a 21 year old student.

Here are the relevant formulas for reference:

[iamcpc]

Can you be more specific about what is wrong with the mathematical solutions to orbit calculations? This is material that would be covered in a 4th year undergraduate Physics course, well within reach of understanding by a 21 year old student.

It's not that they are wrong. Also those formulas you linked were not for systems with 298357 bodies all with mass all pulling on each other. Your orbital speed equation only has two bodies.

Also the gravitational force calculation only has two bodies.

your orbital period could be used to calculate the orbital period of our moon in relation to our earth. Could that same formula used to calculate the orbital period of our moon while it is orbiting our earth which is also orbiting the sun? If so where are the variables for all three of those bodies in that equation?

In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation.[1] The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists,[1] as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.

If by 'mathematical perspective' you mean 'write down an equation that enables us to plug in variables and output position and velocity and data' then no, we can't, but that doesn't mean we can't explain what's going on, or that the planets don't do what we expect them to do. Indeed, we have very accurate predictive models for planetary motion based on the n-body formulae - we just acknowledge that they aren't perfect over long time frames due to both the slight inaccuracies inherent in numerical methods and the sensitivity of the long-term results to small changes in starting conditions, which we can't measure perfectly. If you require that extremely high bar to be reached by every aspect of science, then we would also have no explanation for the behaviour of molecules, atoms and sub-atomic particles, or how aircraft fly, to pick just two examples of things that can't be algebraically solved and require numerical method solutions.

Well that is something that RET can't explain. If we have laws of gravity why can't we put a formula and plug in variables to accurately model the orbits of all these bodies on the solar system?

Questioning it is entirely reasonable; assuming that an entirely baseless alternative model for the shape and disposition of the earth and its solar system, with no supporting evidence, no predictive power, and a whole range of enormous observable contradictions, is in fact a better choice due to some extremely minor limitations at the frontiers of human understanding is just plain daft.

Different people have different views and different levels of strictness on what qualifies as evidence. I assure you that most models are not baseless and each individual person can present you evidence for their own specific FE model.

[SteelyBob]

Can you be more specific about what is wrong with the mathematical solutions to orbit calculations? This is material that would be covered in a 4th year undergraduate Physics course, well within reach of understanding by a 21 year old student.

It's not that they are wrong. Also those formulas you linked were not for systems with 298357 bodies all with mass all pulling on each other. Your orbital speed equation only has two bodies.

Also the gravitational force calculation only has two bodies.

your orbital period could be used to calculate the orbital period of our moon in relation to our earth. Could that same formula used to calculate the orbital period of our moon while it is orbiting our earth which is also orbiting the sun? If so where are the variables for all three of those bodies in that equation?

In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation.[1] The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists,[1] as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.

I refer you to my previous reply. Numerical method solutions do not mean a lack of understanding or plausibility. Also the chaotic nature of the orbits is well understood. Yes, it's chaotic, but over enormous periods of time, meaning we observe a stable situation which is predictable over anything but enormous time spans.

[Tom Bishop]

Numerical solutions tend to be workarounds which are not based on the underlying laws.

https://wiki.tfes.org/Numerical_Solutions

From a question posted on researchgate.net:

  “ Q. What kind of problem solutions do you rate higher: analytical or numerical? More problems can be solved numerically, using computers. But some of the same problems can be solved analytically. What would your preference be? ”

Mohammad Firoz Khan, Ph.D. responds:

  “ A researcher would like to solve it analytically so that it is clear what are premises, assumptions and mathematical rules behind the problem. As such problem is clearly understood. Numerical solution using computers give solution, not the understanding of the problem. It is quite blind. However, in emergency one may resort to this option. ”

Jason Brownlee, Ph.D., tells us on machinelearningmastery.com:

  “ An analytical solution involves framing the problem in a well-understood form and calculating the exact solution. A numerical solution means making guesses at the solution and testing whether the problem is solved well enough to stop. ”

http://www.math.pitt.edu/~sussmanm/2071Spring09/lab02/index.html

  “ With rare exceptions, a numerical solution is always wrong; the important question is, how wrong is it? ”

[SteelyBob]

Numerical solutions tend to be workarounds which are not based on the underlying laws.

https://wiki.tfes.org/Numerical_Solutions

From a question posted on researchgate.net:

  “ Q. What kind of problem solutions do you rate higher: analytical or numerical? More problems can be solved numerically, using computers. But some of the same problems can be solved analytically. What would your preference be? ”

Mohammad Firoz Khan, Ph.D. responds:

  “ A researcher would like to solve it analytically so that it is clear what are premises, assumptions and mathematical rules behind the problem. As such problem is clearly understood. Numerical solution using computers give solution, not the understanding of the problem. It is quite blind. However, in emergency one may resort to this option. ”

Jason Brownlee, Ph.D., tells us on machinelearningmastery.com:

  “ An analytical solution involves framing the problem in a well-understood form and calculating the exact solution. A numerical solution means making guesses at the solution and testing whether the problem is solved well enough to stop. ”

http://www.math.pitt.edu/~sussmanm/2071Spring09/lab02/index.html

  “ With rare exceptions, a numerical solution is always wrong; the important question is, how wrong is it? ”

First of all let's remove Jason Brownlee from the discussion - he is clearly talking about something very different when he refers to 'numerical solutions'. He's isn't wrong in what he's saying, but the problems he is solving are completely different - he is describing using machine learning to search complex problem/solution spaces, hence discussions of techniques like gradient ascent etc, which is indeed a trial-based technique. In numerical methods such as FE in structures, or CFD in aerodynamics, we aren't making guesses and refining by trial and error, we are taking a set of equations that cannot be solved algebraically and applying expertise and judgment to simplify the problem enough to be able to compute an answer that is useful. The methods used to tackle the n-body problem are similar - there's no guesswork involved.

To your opening line:

Numerical solutions tend to be workarounds which are not based on the underlying laws.

You are always demanding evidence. Well, let's see some then. That's a massive statement, and you haven't provided a shred of evidence to support it. Of course, any scientist will always prefer an analytical solution if one is possible, but that doesn't mean that numerical solutions are invalid or indicative of a fallacious theory, as you seem to be trying to suggest. Your last post sums it up quite well - yes, any numerical solution will have a degree of error in it. The skill is in deciding how significant that error is. Given that most aircraft are designed using CFD, we are clearly able to refine results to a usable degree of precision, and the need for CFD does not render the navier stokes equations invalid, nor the underlying theories of aerodynamics.

Evidence then, please, for numerical methods tending to be 'workarounds' and 'not based on the underlying laws'.