Rejection of science

"Calculating the perturbations" is not a legitimate solution to the n-body problems.

Dr. Gopi Krishna Vijaya explains why it is not here:

https://reciprocalsystem.org/PDFa/Replacing%20the%20Foundations%20of%20Astronomy%20(Vijaya,%20Gopi%20Krishna).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:



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 re-derived 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.

~

The Dead End

In the late 19th century, one of the French mathematicians – Henri Poincaré – had already discovered that many of the terms being used in the “perturbation” series by mathematicians like Laplace and Lagrange were becoming infinite for long periods of time, making the system unstable. In simple words, the solutions ‘blow up’ fairly quickly. He also showed that the general problem of 3 mutually gravitating bodies was insoluble through any mathematical analysis! Many physicists and mathematicians built up modern “Chaos theory” based on these ideas, to show simply that one cannot calculate the movements of the planets accurately. Thus began the field of non-linear dynamics.

In the middle of the 20th century, with computers entering the field, the mathematicians pretty much gave up on calculating the orbits by themselves and programmed the computer to do it, even though it was mathematically shown that these orbits were incalculable. They had to be satisfied with approximations or numerical methods (or “brute force” methods.) The result of it all was that after 300 years, Newtonian/Einsteinian thought lands in the same spot that Kepler ended: the orbits point to a living or chaotic system. Only now, there is the additional baggage of all the wrong concepts introduced with regard to “inverse-square law”, “gravitational attraction”, “gravitational mass” and “curved space-time” along with uncountable number of minor assumptions. In this process, an enormous amount of human effort was put to derive thousands of terms in equations over centuries. The entire enterprise has been a wild goose chase

What is more important is the stability of the real Solar System and that can only be investigated numerically as in this paper:

Quote from: Scott Tremaine

Is the solar system stable? by Scott Tremaine, University of Toronto and Institute of Astronomy, Cambridge
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results and implications
The Table summarizes some numerical investigations of the long-term evolution of the solar system. Many follow only the outer five planets (Jupiter to Pluto) since:
(i) the masses of inner planets are so small that the outer planets form an independent dynamical system;
(ii) the large masses of the outer planets suggest that interesting effects are more likely in this region;
(iii) the orbital periods of the outer planets are longer so it is easier to follow the system for a given time.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The maximum timespan over which such calculations may be relevant is 4.5 Gyr backward (the age of the solar system) and 7.7 Gyr forward ( the time until the Sun swallows Mercury and loses a significant portion of its mass; Sackmann et ale 1993). Although
calculations based on secular theory now extend for up to 25 Gyr, the longest N-body integration is only 100 Myr, or 2% of the age of the solar system. Thus the conclusions described below must be treated cautiously. The first important result is that all the planets are still there: none has been ejected, fallen into the Sun, or collided with another planet, and the overall configuration of the planetary system remains quite similar.

Nevertheless, the behaviour of the planets is not boring. Sussman and Wisdom (1988) discovered that the trajectory of Pluto is chaotic: small changes grow exponentially, with an e-folding time (Liapunov time) of 20 Myr. Despite this chaotic behaviour, Pluto's semimajor axis, eccentricity and inclination appeared to vary fairly regularly over the 845 Myr integration. This apparent regularity is impressive, since small disturbances were amplified 10¹⁸ by a factor of exp(⁸⁴⁵/20) ~ over the integration, and suggests that the trajectory is restricted - at least for the timespan of the integration - to a narrow chaotic zone in phase space.

That dates back to 1994 and there are far faster computers available now.

But look at the time-scales! Scott Tremaine is talking about time-scales of Myr and Gyr!

So we cannot prove that the Solar System is stable and it presumably is not over such long periods but who cares?

Now Tom, please change the record! You three-body problem is evidence of nothing of importance and has worn thin!

That paper says it's using Symplectic integrators.

From the paper:



We then see this paper on symplectic integrators:

Computing the long term evolution of the solar system with geometric numerical integrators

https://imaginary.org/snapshot/computing-the-long-term-evolution-of-the-solar-system-with-geometric-numerical-integrators

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 so-called geometric numerical methods. Short sample codes are provided for the Sun–Earth–Moon system.

...

Numerical methods: For a general differential equation, however, it is in practice often difficult or even impossible to find a formula for the exact solution. Therefore a numerical integrator, which is an algorithmic method for calculating an approximate solution, must be used.

....





...

In 1885, King Oscar II of Sweden sponsored a competition about this question.
The prize was awarded to Henri Poincaré (1854–1912), although he did not
really solve the problem. His contribution, however, is at the origin of the theory
of dynamical systems. It also led to important developments in “Hamiltonian
perturbation theory” and gave rise to the so-called Kolmogorov–Arnold–Moser
(KAM) theory, which deals with the persistence of quasi-periodic motions under
small perturbations, see the survey [7]. Unfortunately, this beautiful theory
does not apply to realistic solar system models.

...

We have seen that the energy, a key invariant of all mechanical systems, is
well preserved by the symplectic Euler method. In contrast, the explicit Euler
method, and more generally any standard explicit Runge–Kutta methods, do
not preserve it and are thus not suitable for integration over long time intervals.
A mathematical theory called “backward error analysis” permits to demonstrate
that symplectic integrators have a good energy conservation for such mechanical
systems.

A special integrator which preserves the energy and prevents the body from escaping is used. Does this sound like a full clean simulation of gravity?

"Calculating the perturbations" is not a legitimate solution to the n-body problems.

Dr. Gopi Krishna Vijaya explains why it is not here:

https://reciprocalsystem.org/PDFa/Replacing%20the%20Foundations%20of%20Astronomy%20(Vijaya,%20Gopi%20Krishna).pdf

That's quite irrelevant because perturbations are useful only used for short term estimates and the method used now is numerical integration.

You have read Rejection of science « Reply #54 on: Today at 08:11:46 AM »

But you go on and on ad nauseum on the stability of Solar System. Would you care to prove that your "flat Earth Cosmology" is even possibility.

Sun
The Sun is a revolving sphere. It has a diameter of 32 miles and is located approximately 3000 miles above the surface of the Earth.

Moon
The Moon is a revolving sphere. It has a diameter of 32 miles and is located approximately 3000 miles above the surface of the earth.

And

Planets
The planets are spherical bodies which revolve above the Earth. The planets follow a similar daily route across the sky as the Sun. Five planets — Mercury, Venus, Mars, Jupiter, and Saturn are visible to the naked eye and were known to the ancients as "wandering stars;" entities which appear to move differently from the fixed path of the stars.

Please tell us what keeps those bodies in their orbits so precisely that their positions can be predicted years ahead.

The Sun and Moon, in particular, have to periodically change their velocities.
The Sun goes through this cycle from circling:
        over Equator to over the Tropic of Cancer, then back
        over the Equator to over the Tropic of Capricorn  (it's close to there now) and finally to
        over Equator in one year.
And the Moon covers a larger range of latitudes but does in each Lunar month.

How are these motions controlled on your flat Earth? At least Albert Smith (Zetetes) had the honesty to claim "doubtless under intelligent supervision" !
.

"It (the Sun) then goes round with the southern currents, daily, contracting its circle in a fine spiral until it arrives at 23¹/₂°S. when, having lost its further southern tendency or swirl, electrical and magnetic forces, doubtless under intelligent supervision, drive it again northwards. Similar explanations apply to the moon, and to the planets, but with different periods, owing to their different altitudes, as already explained in a former article."

He is literally quoting back what has already been put to him in this thread, the whole argument is based on the lack of a simple elegant equation. And quoting RE sources, again, in an attempt to presumably prove FE??

The error factors are incredibly small, and more than capable of predicting multiple bodies in a system for deep space navigation.

https://www.bbc.co.uk/news/science-environment-20033940


And what do you know it worked;

 https://www.planetary.org/blogs/guest-blogs/2013/20130926-gravity-assist.html

It worked with enough confidence for valuable payload being given over for interstellar space experiments.

"Calculating the perturbations" is not a legitimate solution to the n-body problems.

Dr. Gopi Krishna Vijaya explains why it is not here:

https://reciprocalsystem.org/PDFa/Replacing%20the%20Foundations%20of%20Astronomy%20(Vijaya,%20Gopi%20Krishna).pdf

Quote

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:



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 re-derived 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.

~

The Dead End

In the late 19th century, one of the French mathematicians – Henri Poincaré – had already discovered that many of the terms being used in the “perturbation” series by mathematicians like Laplace and Lagrange were becoming infinite for long periods of time, making the system unstable. In simple words, the solutions ‘blow up’ fairly quickly. He also showed that the general problem of 3 mutually gravitating bodies was insoluble through any mathematical analysis! Many physicists and mathematicians built up modern “Chaos theory” based on these ideas, to show simply that one cannot calculate the movements of the planets accurately. Thus began the field of non-linear dynamics.

In the middle of the 20th century, with computers entering the field, the mathematicians pretty much gave up on calculating the orbits by themselves and programmed the computer to do it, even though it was mathematically shown that these orbits were incalculable. They had to be satisfied with approximations or numerical methods (or “brute force” methods.) The result of it all was that after 300 years, Newtonian/Einsteinian thought lands in the same spot that Kepler ended: the orbits point to a living or chaotic system. Only now, there is the additional baggage of all the wrong concepts introduced with regard to “inverse-square law”, “gravitational attraction”, “gravitational mass” and “curved space-time” along with uncountable number of minor assumptions. In this process, an enormous amount of human effort was put to derive thousands of terms in equations over centuries. The entire enterprise has been a wild goose chase

Reciprocalsysyem.org?

I thought you were insisting on mainstream science, not fringe theories?  3 dimensions of time?  I’ve not heard that one before.

And no.  Perturbations are not like epicycles.  Epicycles were a fudge because there was no known reason for planets to have little orbits around nothing.  Perturbation of orbits are based on the same gravitational attraction as the orbits themselves and are derived from those equations, neatly fitting with observational evidence.

You link complains that computers are used for this, but why?  As I said a moment ago, its simply a practical way to perform the thousands of calculations necessary that would be complete waste of time for huge teams of very highly qualified mathematicians to do manually.

Saying that mathematicians gave up and let computers do it for them is stupid.  The programs are still written by physicists and mathematicians.  They just let computers do the donkey work.

Quote from: rabinoz on December 24, 2019, 02:11:46 PM

What is more important is the stability of the real Solar System and that can only be investigated numerically as in this paper:
Is the solar system stable? by Scott Tremaine, University of Toronto and Institute of Astronomy, Cambridge

That dates back to 1994 and there are far faster computers available now.

But look at the time-scales! Scott Tremaine is talking about time-scales of Myr and Gyr!

So we cannot prove that the Solar System is stable and it presumably is not over such long periods but who cares?

Now Tom, please change the record! You three-body problem is evidence of nothing of importance and has worn thin!

That paper says it's using Symplectic integrators. "Symplectic integrators"

Sure and there is a very sound reason for that and that is their inherent energy preserving properties.

From the paper:



We then see this paper on symplectic integrators:

Quote from: Shaula Fiorelli Vilmart and Gilles Vilmar

Computing the long term evolution of the solar system with geometric numerical integrators
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 so-called geometric numerical methods. Short sample codes are provided for the Sun–Earth–Moon system.
...
Numerical methods: For a general differential equation, however, it is in practice often difficult or even impossible to find a formula for the exact solution. Therefore a numerical integrator, which is an algorithmic method for calculating an approximate solution, must be used.
....

That is using numerical integration to solve for the motion of an undamped mass on a spring.
Can't you see at a glance that both the explicit and implicit Euler methods rapidly accumulate errors unless an extremely small step size is used.
One the other hand the symplectic integrator can give good results for the same step size.

And the Moon-Earth-Sun demonstrates  the same problem for the simple Euler methods

Quote from: Shaula Fiorelli Vilmart and Gilles Vilmar

In 1885, King Oscar II of Sweden sponsored a competition about this question.
The prize was awarded to Henri Poincaré (1854–1912), although he did not really solve the problem. His contribution, however, is at the origin of the theory of dynamical systems. It also led to important developments in “Hamiltonian perturbation theory” and gave rise to the so-called Kolmogorov–Arnold–Moser (KAM) theory, which deals with the persistence of quasi-periodic motions under small perturbations, see the survey [7] ]. Unfortunately, this beautiful theory
does not apply to realistic solar system models.

The initial question “Is the solar system stable?” then remained open until the last decades, where the final negative answer, revealing that the solar system is chaotic, was given by the mathematician and astronomer Jacques Laskar and his collaborators. The argument is based on analytic means, but also uses numerical methods including geometric integrators.

In addition, some of their recent computations show that collisions or ejections could even occur in the next five billion years, that is, before the end of the life of the Sun.

I added a bit and note again the time-scale, "in the next five billion years".

Quote from: Shaula Fiorelli Vilmart and Gilles Vilmar

We have seen that the energy, a key invariant of all mechanical systems, is well preserved by the symplectic Euler method. In contrast, the explicit Euler method, and more generally any standard explicit Runge–Kutta methods, do not preserve it and are thus not suitable for integration over long time intervals.

A mathematical theory called “backward error analysis” permits to demonstrate that symplectic integrators have a good energy conservation for such mechanical systems.

A special integrator which preserves the energy and prevents the body from escaping is used. Does this sound like a full clean simulation of gravity?

"A special integrator which preserves the energy" sounds extremely valuable because any accurate numerical analysis of such a system must "preserve the energy" or the results is simply incorrect!

You seem to have no real understanding of a system like this.

Preserving the energy does not prevent any body from escaping and so is quite unrelated to "a full clean simulation of gravity".

Isn't it obvious to you that unless there is energy in or out from outside that the total energy in the system must remain unchanged? .

I see that you continue to post nonsense with only yourself as the source. Recall that you should debate as if your opinion is trash unless backed by a source.

Here is a source which says that symplectic integrators always give stable conditions regardless of the perturbations which affects the body:

Numerical Integration Techniques in Orbital Mechanics Applications

https://www.researchgate.net/publication/281108779_Numerical_Integration_Techniques_in_Orbital_Mechanics_Applications

Revisiting Table I, the non-symplectic integrators do not give a stable solution even at time step of 1e-4. The solutions from these integrators are chaotic and shoot off to infinity.

...

The symplectic integrators are very good at keeping the orbit stable for even long period integrations. For all the integrators that provide stable solution at time step of 1e-3, increasing the integration time to six orbital period yields the same result. Symplectic integrators are particularly good at this since it can keep the errors bounded.

...

From the two problems analyzed in this paper, one can clearly see the advantages of the symplectic integrator over the non-symplectic integrators. The symplectic integrators produce consistently better results with higher accuracy and slightly less run time. The fourth order symplectic integrators in particular are extremely successful in propagating both the restricted three body problem and the simple two body problem. As discussed earlier, the symplectic integrator is able to achieve stable solutions at lower integration tolerance and run time. For long duration integration, which is often the case for most orbital mechanics applications, the error on the non-symplectic integrators are unbounded and can grow. On the other hand, the symplectic integrator, by keeping the Hamiltonian constant, is able to bound the error and prevent it from growing substantially, always able to return to stable condition after perturbations.

The non-sympletic integrators are bad because they are chaotic, while the 'accurate' symplectic integrators are "always able to return to stable conditions after perturbations".

Clearly, one type of integrator reflects the explicit rules of the system and the other does not.

I see that you continue to post nonsense with only yourself as the source. Recall that you should debate as if your opinion is trash unless backed by a source.

Keep your insults to yourself, Mr Bishop. But where was my opinion not backed by a reputable source?

I am no authority on numerical integration but did a little on not too dissimilar material in the electronic circuit simulation programs, SINC and SPICE, while at UCB in 1972/3.
SPICE was written by Laurence Nagel for his Ph.D and released in 1973 but I was't smart enough to write that sort of thing but did present a seminar on the different integration methods. So I'm quite aware of the integration errors from the simple Euler integration etc. Back there SPICE used what they called "trapezoidal integration" - sort of a combined of the two Euler methods. But it's a bit hard remembering details from 46 years ago.

Here is a source which says that symplectic integrators always give stable conditions regardless of the perturbations which affects the body:

Please point out where exactly it states that "symplectic integrators always give stable conditions regardless of the perturbations which affects the body".

It does say this:

the symplectic integrator, by keeping the Hamiltonian constant, is able to bound the error and prevent it from growing substantially[/i], always able to return to stable condition after perturbations.

The integration method should be stable but that does not imply that the solution is stable.

Look again at these diagrams from your earlier reference:
      Computing the long term evolution of the solar system with geometric numerical integrators

Both the explicit Euler and the implicit Euler clearly have massive errors but the symplectic integrator remains bounded exactly as it should for the same step size.

Then for the Sun-Earth-Moon system.

Note how for the same step size the symplectic integrator shows the solutions expected but again the explicit Euler method goes wild well before a tear has elapsed.

Quote from: Evan Anzalone & Patrick Chai

Numerical Integration Techniques in Orbital Mechanics Applications
Revisiting Table I, the non-symplectic integrators do not give a stable solution even at time step of 1e-4. The solutions from these integrators are chaotic and shoot off to infinity.
...
The symplectic integrators are very good at keeping the orbit stable for even long period integrations. For all the integrators that provide stable solution at time step of 1e-3, increasing the integration time to six orbital period yields the same result. Symplectic integrators are particularly good at this since it can keep the errors bounded.
...
From the two problems analyzed in this paper, one can clearly see the advantages of the symplectic integrator over the non-symplectic integrators. The symplectic integrators produce consistently better results with higher accuracy and slightly less run time. The fourth order symplectic integrators in particular are extremely successful in propagating both the restricted three body problem and the simple two body problem. As discussed earlier, the symplectic integrator is able to achieve stable solutions at lower integration tolerance and run time. For long duration integration, which is often the case for most orbital mechanics applications, the error on the non-symplectic integrators are unbounded and can grow. On the other hand, the symplectic integrator, by keeping the Hamiltonian constant, is able to bound the error and prevent it from growing substantially, always able to return to stable condition after perturbations.
This is particularly visible in the two body problem example discussed earlier, where the error of the non-symplectic integrator is 7500 times the fourth order symplectic.

The non-sympletic integrators are bad because they are chaotic, while the 'accurate' symplectic integrators are "always able to return to stable conditions after perturbations".

I added a bit to your quote, "in the two body problem example discussed earlier, where the error of the non-symplectic integrator is 7500 times the fourth order symplectic."

Clearly, one type of integrator reflects the explicit rules of the system and the other does not.

Yes and two of those explicit rules are that the system energy must be conceived and the  system angular momentum must be conceived.

In the words of the author, "From the two problems analyzed in this paper, one can clearly see the advantages of the symplectic integrator over the non-symplectic integrators. The symplectic integrators produce consistently better results with higher accuracy and slightly less run time."

Look!
A numerical integration is wrong if it gives chaotic solutions where the real system does not.
And the explicit and implicit Euler methods do just that unless an very small step size is used and then computational errors will grow.

You quote these bits of papers but have seem to have no real understanding of their meaning.

Wow did you catch him partially quoting again 

Convinced hes just a troll

University of Rochester
PHY411 Lecture notes Part 7 – Integrators

http://astro.pas.rochester.edu/~aquillen/phy411/lecture7.pdf



Take a guess at which one is symplectic.

Further down in the document:



...



Yet again we see that your opinion is unable to be demonstrated. It is stated directly that the purpose of the integrator is to preserve the area or geometry of the phase space.

Tom have you been taking lessons from Sandy?  Sometimes you don't need to resort to mathematics to see that something works.  You just have to look up to the sky and watch what is going on up there. Watch the behaviour of the Moon and the other bright planets during the year. The movements they show is clear and direct evidence that the Moon orbits the Earth and the Earth, along with the other planets orbit the Sun.

You always have to complicate things more than you need to in order to try and 'prove' that what you believe in is right and everyone (or shall we say by everyone I mean RE) else is wrong.  I have got used to you ignoring everything I say but you do the same to anyone who tries to knock some sense into you..

Why should we believe in RET when its proponents are unable to do a Google search and find a single quote showing that it is possible for a planet with a moon to revolve around a star?

Because all we have to counter that are FEers spouting pathetic lies.

No physicist has ever said that such configurations cannot exist.
You are yet to provide a single quote.
Stop spamming links to your site and provide a valid citation from mainstream physics that claims such configurations cannot exist.

The fact that the scientific model has the such systems it is quite clear that mainstream physicists do believe such a system can exist.

Why believe something that can't even be simulated by the collective efforts of the greatest mathematicians in history?

And there you go confusing words again.
It can be simulated, and is done so quite often. It cannot be "solved" as in have a nice simple solution for it.

The 2 are not the same thing, stop pretending they are.

You should debate by assuming that your own opinion is trash. If all you can give us is trash, then it will be discarded as trash.

So why don't you ever do that? Instead you come in acting like your own opinion and misrepresentations are irrefutable facts.
Why don't you debate by assuming that your own opinion is pure garbage and try finding a single quote to back up your baseless claims?

Why don't you try finding a single mainstream physicist that says that your pizza planet with a sun and moon held up by pure magic is possible?

Or is this your way of telling everyone that you are actually a REer?

University of Rochester
PHY411 Lecture notes Part 7 – Integrators

http://astro.pas.rochester.edu/~aquillen/phy411/lecture7.pdf



Take a guess at which one is symplectic.

Easy, the one on the left that conserves energy as is necessary in any accurate solution!
Read and understand your own reference!
"In the middle we show a system where volume contracts, as would occur when energy dissipation takes place" and that's not permitted in a conservative system and
"On the right the system gains energy" and again that's not permitted in a conservative system

Further down in the document:


...


Yet again we see that your opinion is unable to be demonstrated. It is stated directly that the purpose of the integrator is to preserve the area or geometry of the phase space.

No we don't.
I have not been stating my opinion but I'm certain that on this topic my opinion would be worth far more than yours.
All we see is that your failing to understand your own references.

Yes, "It is stated directly that the purpose of the integrator is to preserve the area or geometry of the phase space."
But that is phase space not the physical space and the area in phase space is not directly related to the locations of the planets etc.

Please, in future do not quote references that you do not understand.

Or is this your way of telling everyone that you are actually a REer?

That's my conclusion,  if you go through Tom's posts he only ever puts forward RE evidence, the wiki is full of it. Despite being asked on numerous occasions he cant produce anything even half way FE.

Hes an RE troll who thinks hes more clever than he actually is, a Sandy alt. But I suspect Sandy does it as he wants to be recognised as some kind of Maths guru, Tom? Looks like ego stroking.

With it being Christmas morning etc, I missed this one.

Why should we believe in RET when its proponents are unable to do a Google search and find a single quote showing that it is possible for a planet with a moon to revolve around a star? Physicists say that 400 years of research has shown that only very special symmetrical configurations can exist.

No they haven't! They might have shown that about analytic solutions to the-three-body problem but that's quite a different matter.
 

It has already been proven dynamically that the system imagined by Copernicus is unfeasible.

• And where has that been proven?
  
  
• But, so what? The Solar System does not follow the "system imagined by Copernicus" anyway so who cares.

Why believe something that can't even be simulated by the collective efforts of the greatest mathematicians in history?

• Because it has been observed in real life for centuries and reality beats simulations hands down any day!
  
  
• And because your claim it totally false anyway. It may have been true 50 years ago but not any more. Read this again!
  The Solar System can now be analysed numerically as in this paper:
  
  Quote from: Scott Tremaine

  Is the solar system stable? by Scott Tremaine, University of Toronto and Institute of Astronomy, Cambridge
  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
  Results and implications
  The Table summarizes some numerical investigations of the long-term evolution of the solar system. Many follow only the outer five planets (Jupiter to Pluto) since:
  (i) the masses of inner planets are so small that the outer planets form an independent dynamical system;
  (ii) the large masses of the outer planets suggest that interesting effects are more likely in this region;
  (iii) the orbital periods of the outer planets are longer so it is easier to follow the system for a given time.
  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
  The maximum timespan over which such calculations may be relevant is 4.5 Gyr backward (the age of the solar system) and 7.7 Gyr forward ( the time until the Sun swallows Mercury and loses a significant portion of its mass; Sackmann et ale 1993). Although
  calculations based on secular theory now extend for up to 25 Gyr, the longest N-body integration is only 100 Myr, or 2% of the age of the solar system. Thus the conclusions described below must be treated cautiously. The first important result is that all the planets are still there: none has been ejected, fallen into the Sun, or collided with another planet, and the overall configuration of the planetary system remains quite similar.
  
  Nevertheless, the behaviour of the planets is not boring. Sussman and Wisdom (1988) discovered that the trajectory of Pluto is chaotic: small changes grow exponentially, with an e-folding time (Liapunov time) of 20 Myr. Despite this chaotic behaviour, Pluto's semimajor axis, eccentricity and inclination appeared to vary fairly regularly over the 845 Myr integration. This apparent regularity is impressive, since small disturbances were amplified 10¹⁸ by a factor of exp(⁸⁴⁵/20) ~ over the integration, and suggests that the trajectory is restricted - at least for the timespan of the integration - to a narrow chaotic zone in phase space.
  

  That dates back to 1994 and there are far faster computers available now.

There are things that exist regrdless of the ability of modern math to describe them.

Square with the same area as a given circle won't stop existing if we just can't construct it.
Three or more bodies won't stop moving just because we don't always know how to calculate their movement.

But that is phase space not the physical space and the area in phase space is not directly related to the locations of the planets etc.

You need to update your studies of bifurcation theory.

Here is one of the foremost experts of all time in bifurcation theory, John Guckenheimer, explaining what a phase space is:

http://chaosbook.org/chapters/flows.pdf

All possible values for positions and velocities of the planets form the phase space of the system.

Please, in future do not quote references that you do not understand.

Right.


Now,  until 1991 the only available numerical integration of a realistic model of the full heliocentric solar system could be used for an interval of 44 centuries (4,400 years).

That is why J. Laskar replaced the full Newtonian equations of the motion by the so-called secular system introduced by Lagrange where the fast angular variables are eliminated. This system, instead of giving the fast motion of the planets in space, describes the slow deformation of the planets’ orbits.

The astrophysicists thought now that everything is fine.

However, everything turned out to be wrong.

Dr. Robert W. Bass

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

Dr. Bass' basic discovery:

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

That is why, excluding the Lyapunov exponents which may have no relation to the true exponent and the crucial sensitivity of the initial conditions problem, the interval of assured reliability for Newton's equations of gravitational motion is at most three hundred years.

All of the details, including an analysis of the theorems of Nekhoroshev and Saari:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1935048#msg1935048

What does this mean? It means that an approach using NONLINEAR DIFFERENTIAL EQUATIONS cannot be used to describe the heliocentrical solar system.

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

The exponential divergence of chaotic trajectories precludes long-term prediction given the limited knowledge of the state of our solar system.

Quote from: Tom Bishop on December 25, 2019, 12:50:26 AM

University of Rochester
PHY411 Lecture notes Part 7 – Integrators

http://astro.pas.rochester.edu/~aquillen/phy411/lecture7.pdf



Take a guess at which one is symplectic.

Easy, the one on the left that conserves energy as is necessary in any accurate solution!
Read and understand your own reference!
"In the middle we show a system where volume contracts, as would occur when energy dissipation takes place" and that's not permitted in a conservative system and
"On the right the system gains energy" and again that's not permitted in a conservative system

Quote from: Tom Bishop

Further down in the document:


...


Yet again we see that your opinion is unable to be demonstrated. It is stated directly that the purpose of the integrator is to preserve the area or geometry of the phase space.

No we don't.
I have not been stating my i]opinion[/i] but I'm certain that on this topic my opinion would be worth far more than yours.
All we see is that your failing to understand your own references.

Yes, "It is stated directly that the purpose of the integrator is to preserve the area or geometry of the phase space."
But that is phase space not the physical space and the area in phase space is not directly related to the locations of the planets etc.

Please, in future do not quote references that you do not understand.

Again, you are posting your opinions. You have not backed up any of your interpretations with sources.

You could easily look up the definition of "phase space" and back up your own personal definition that you apparently made up on the spot.

https://en.m.wikipedia.org/wiki/Phase_space

"In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space."

It says nothing about phase space being unrelated to points in normal space.

You are unable to justify what you think and feel what is occuring with any sources. You debate by expecting us to accept you as expert, when you continuously fail to support yourself.

"Easy, the one on the left conserves energy as I. Any accurate solution!" -- again, nonsense. You have not demonstrated this opinion. Why should it not be possible for the orbit to lose or gain energy through influences like the trajectories which spiral inwards or outwards in the diagram? You are posting interpretation with nothing to support you. Kindly demonstrate what you think is occuring with a source which is not your own self.

Quote from: JackBlack on December 25, 2019, 01:07:40 AM

Or is this your way of telling everyone that you are actually a REer?

That's my conclusion,  if you go through Tom's posts he only ever puts forward RE evidence, the wiki is full of it. Despite being asked on numerous occasions he cant produce anything even half way FE.

Anything that contradicts RE often supports FE.

Galaxies which are supposedly millions of light years away from each other move in synchronized patterns that cannot be explained, contradicting the concept of a large universe.

https://futurism.com/the-byte/something-strange-synchronizing-distant-galaxies

Something Very Strange Seems to Be Synchronizing Distant Galaxies

  “ Galaxies millions of light years away seem to be connected by an unseen network of massive intergalactic structures, which force them to synchronize in ways that can’t be explained by existing astrophysics, Vice reports. The discoveries could force us to rethink our fundamental understanding of the universe.

“The observed coherence must have some relationship with large-scale structures, because it is impossible that the galaxies separated by six megaparsecs [roughly 20 million light years] directly interact with each other,” Korea Astronomy and Space Science Institute astronomer Hyeop Lee told the site.

In Sync

There have been many instances of astronomers observing galaxies that seem to be connected and moving in sync with each other. A study by Lee, published in The Astrophysical Journal in October, found that hundreds of galaxies are rotating in exactly the same way, despite being millions of light years apart.

And a separate study, published in the journal Astronomy and Astrophysics in 2014, found supermassive black holes aligning with each other, despite being billions of light years apart.

Spooky Action

While current cosmological principles support the alignment and movement of ancient stars at a smaller scale, astronomers are puzzled by the much, much larger patterns across vast distances.

But before they can draw any conclusions, they’ll need more data — the body of work is still limited, as Vice points out. ”

This clearly contradicts the RE concept of a large universe, while also supporting the FE small universe. It is stated that it cannot be explained by existing physics.

Yet, when confronted with things like this we have people posting one liners on an online forum with zero supporting sources, thinking that they personally "explained" or "solved" great problems in Astrophysics.

Never stopped a flat earther.

“ But before they can draw any conclusions, they’ll need more data — the body of work is still limited, as Vice points out.”


Also not pictured, the Hubble space telescope.

Again, you are posting your opinions. You have not backed up any of your interpretations with sources.

No, that would be you.
So far all you have done is post your opinions. Your sources do not back up your opinions at all.

We are still waiting on a source which says 3 body systems cannot exist.

It says nothing about phase space being unrelated to points in normal space.

And who said anything like that?
Rab said that the area in phase space is not directly related to the area in real space.
And guess what? That matches your quote as well.
Just look a the first image provided on that page. What are the axes used? x and dx/dt.
It is describing an oscillator which moves only in 1 D.
The area in real space is thus 0 as it is a line. But the area in phase space is not.

You are unable to justify what you think and feel what is occuring with any sources. You debate by expecting us to accept you as expert, when you continuously fail to support yourself.

Good job projecting again.

You are yet to provide any valid sources to justify your claims; instead you provide sources with a tangential link and pretend they justify your claims.

And no, articles from magazines, which often get the science completely wrong, are not valid sources, especially when they don't actually back you up.
Your futurism article provides nothing to indicate that the FE model is correct. All it does is claim there is something that cannot be explained yet.

Quote from: mak3m on December 25, 2019, 03:18:57 AM

Quote from: JackBlack on December 25, 2019, 01:07:40 AM

Or is this your way of telling everyone that you are actually a REer?

That's my conclusion,  if you go through Tom's posts he only ever puts forward RE evidence, the wiki is full of it. Despite being asked on numerous occasions he cant produce anything even half way FE.

Anything that contradicts RE often supports FE.

Galaxies which are supposedly millions of light years away from each other move in synchronized patterns that cannot be explained, contradicting the concept of a large universe.

https://futurism.com/the-byte/something-strange-synchronizing-distant-galaxies

Something Very Strange Seems to Be Synchronizing Distant Galaxies

  “ Galaxies millions of light years away seem to be connected by an unseen network of massive intergalactic structures, which force them to synchronize in ways that can’t be explained by existing astrophysics, Vice reports. The discoveries could force us to rethink our fundamental understanding of the universe.

“The observed coherence must have some relationship with large-scale structures, because it is impossible that the galaxies separated by six megaparsecs [roughly 20 million light years] directly interact with each other,” Korea Astronomy and Space Science Institute astronomer Hyeop Lee told the site.

In Sync

There have been many instances of astronomers observing galaxies that seem to be connected and moving in sync with each other. A study by Lee, published in The Astrophysical Journal in October, found that hundreds of galaxies are rotating in exactly the same way, despite being millions of light years apart.

And a separate study, published in the journal Astronomy and Astrophysics in 2014, found supermassive black holes aligning with each other, despite being billions of light years apart.

Spooky Action

While current cosmological principles support the alignment and movement of ancient stars at a smaller scale, astronomers are puzzled by the much, much larger patterns across vast distances.

But before they can draw any conclusions, they’ll need more data — the body of work is still limited, as Vice points out. ”

This clearly contradicts the RE concept of a large universe, while also supporting the FE small universe. It is stated that it cannot be explained by existing physics.

Yet, when confronted with things like this we have people posting one liners on an online forum with zero supporting sources, thinking that they personally "explained" or "solved" great problems in Astrophysics.

That's not how it works, drifting around the fringes of science  quoting known anomalies, and always over stating them does not prove FE.

You have been asked countless times to produce FE proof, any observation, predictions or a working model or map, and you cant.

RE needs more data doesnt equal flat earth.

I have provided you links on a practical example of the n body problem and the orbital mechanics for the voyager missions. The calculations were incredibly accurate.

You keep ducking and diving those questions though

They say that they have made observations of hundreds of galaxies that do this. They need more observations in hopes of explaining it, not that they did not make the observations, which are described as unexplained. A pretty weak response.

They say that they have made observations of hundreds of galaxies that do this.

Which in no way indicates that these galaxies are close or that Earth is flat.

A problem in Astronomy doesn't magically mean Earth is flat, nor is it evidence that Earth is flat.
It doesn't negate the fact that RE actually has explanations for lots of things, while FE does not.

Quote from: Tom Bishop on December 25, 2019, 08:29:56 AM

Again, you are posting your opinions. You have not backed up any of your interpretations with sources.

No, that would be you.
So far all you have done is post your opinions. Your sources do not back up your opinions at all.

We are still waiting on a source which says 3 body systems cannot exist.

A whole page of sources was given to you. Caltech physicist Sean Carrol says that the three body problem was proven to be unstable.

http://www.preposterousuniverse.com/blog/2006/07/23/n-bodies/

  “ For N=2 the complete set of solutions is straightforward and has been known for a long time — each body moves in a conic section (circle, ellipse, parabola or hyperbola) around the center of mass. In fact, Kepler found the solution even before Newton came up with the problem!

But let N=3 and chaos breaks loose, quite literally. For a long time people recognized that the motion of three gravitating bodies would be a difficult problem, but there were hopes to at least characterize the kinds of solutions that might exist

~

In 1885, a mathematical competition was announced in honor of the 60th birthday of King Oscar II of Sweden, and the three-body problem was one of the questions. ...Henri Poincare was a favorite to win the prize, and he submitted an essay that demonstrated the stability of planetary motions in the three-body problem (actually the “restricted” problem, in which one test body moves in the gravitational field generated by two others). In other words, without knowing the exact solutions, we could at least be confident that the orbits wouldn’t go crazy; more technically, solutions starting with very similar initial conditions would give very similar orbits. Poincare’s work was hailed as brilliant, and he was awarded the prize.

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

He goes on to state that only special highly symmetrical orbits may exist.

Quote from: Tom Bishop on December 25, 2019, 08:29:56 AM

It says nothing about phase space being unrelated to points in normal space.

And who said anything like that?
Rab said that the area in phase space is not directly related to the area in real space.
And guess what? That matches your quote as well.
Just look a the first image provided on that page. What are the axes used? x and dx/dt.
It is describing an oscillator which moves only in 1 D.
The area in real space is thus 0 as it is a line. But the area in phase space is not.

Nonsense. It does not state that at all. You have provided no source.

Quote from: mak3m on December 25, 2019, 03:18:57 AM

Quote from: JackBlack on December 25, 2019, 01:07:40 AM

Or is this your way of telling everyone that you are actually a REer?

That's my conclusion,  if you go through Tom's posts he only ever puts forward RE evidence, the wiki is full of it. Despite being asked on numerous occasions he cant produce anything even half way FE.

Anything that contradicts RE often supports FE.

If that "anything" really did "contradict RE" it might "support FE" but it doesn't.

Galaxies which are supposedly millions of light years away from each other move in synchronized patterns that cannot be explained, contradicting the concept of a large universe.

That might raise questions about modern Cosmology but is totally irrelevant to the question of whether the Earth is flat or near-spherical.

The spherical-Earth was accepted and fitted available evidence fairly closely for about 1800 years before the time of Copernicus.
Many people did recognise that there were problems.

• Aristarchus of Samos considered it illogical that a huge object, the Sun, should orbit a much smaller object, the Earth and many others raised similar questions.
  You might read Muslim Heritage: Copernicus and Arabic Astronomy: A Review of Recent Research by George Saliba
  and The Solar and Lunar Theory of Ibn ash-Shatir, A Pre-Copernican Copernican Model by Victor Robert
  
  
• Then astronomical observations, such as the retrograde motion of Mars etc, required such complicated explanations in Ptolemy's model that Copernicus tried to explain it better with his heliocentric system.
  
  But he was still hampered by the old ideas that the orbits of planets must be perfect circles and so his model was no better that that of Ptolemy.
  
  But finally, after Tycho Brahe's work, Kepler postulated that the planetary should be elliptical but he had no theory of the ellipse to work with so had to work with geometric curve fitting to data that still had some position errors.
  So it wasn't until after Newton that the Heliocentric Solar System was put on a firm footing.
  
  Nevertheless, we now know that the Kepler model is still only an approximation though quite a ggod because the Sun's mass is so large compared to the mass of the rest of the Solar System.
  

But ant perceived discrepancies in the motions of "Galaxies which are supposedly millions of light-years away" does not raise the slight question about the validity of none of the Earth being a Globe or even of the Heliocentric Solar System.

You said that:

Anything that contradicts RE often supports FE.

I presume that if there are similar discrepancies in FET then we can take that as evidence that the Earth is a Globe.

Well, what about this:

Rowbotham claimed that the Sun could no be higher than 700 statute miles.
Most recent flat Earthers seem to claim "about 3100 miles".
Flat Earth Scientist, Sandokhan clams something like 10 miles and
I believe I've seen you claim about 6100 miles.

That looks to be a massive weakness.

---

Then you seem to have forgotten all about this:
But you go on and on ad nauseum on the stability of Solar System. Would you care to prove that your "flat Earth Cosmology" is even a possibility.

Sun
The Sun is a revolving sphere. It has a diameter of 32 miles and is located approximately 3000 miles above the surface of the Earth.

Moon
The Moon is a revolving sphere. It has a diameter of 32 miles and is located approximately 3000 miles above the surface of the earth.

And

Planets
The planets are spherical bodies which revolve above the Earth. The planets follow a similar daily route across the sky as the Sun. Five planets — Mercury, Venus, Mars, Jupiter, and Saturn are visible to the naked eye and were known to the ancients as "wandering stars;" entities which appear to move differently from the fixed path of the stars.

Please tell us what keeps those bodies in their orbits so precisely that their positions can be predicted years ahead.

I thought that you claimed that a three-body system must be unstable.

The Sun and Moon, in particular, have to periodically change their velocities.
The Sun goes through this cycle from circling:
        over Equator to over the Tropic of Cancer, then back
        over the Equator to over the Tropic of Capricorn  (it's close to there now) and finally to
        over Equator in one year.
And the Moon covers a larger range of latitudes but does in each Lunar month.

How are these motions controlled on your flat Earth? At least Albert Smith (Zetetes) had the honesty to claim "doubtless under intelligent supervision" !
.

"It (the Sun) then goes round with the southern currents, daily, contracting its circle in a fine spiral until it arrives at 23¹/₂°S. when, having lost its further southern tendency or swirl, electrical and magnetic forces, doubtless under intelligent supervision, drive it again northwards. Similar explanations apply to the moon, and to the planets, but with different periods, owing to their different altitudes, as already explained in a former article."

This works two ways and I see these total failures of FET as evidence that the Earth is a Globe.

So, some answers from you, please!

Quote from: JackBlack on December 25, 2019, 12:17:52 PM

And who said anything like that?
Rab said that the area in phase space is not directly related to the area in real space.
And guess what? That matches your quote as well.
Just look a the first image provided on that page. What are the axes used? x and dx/dt.
It is describing an oscillator which moves only in 1 D.
The area in real space is thus 0 as it is a line. But the area in phase space is not.

Nonsense. It does not state that at all. You have provided no source.

Have you forgotten your own reference? Read again!

We then see this paper on symplectic integrators:
Computing the long term evolution of the solar system with geometric numerical integrators

https://imaginary.org/snapshot/computing-the-long-term-evolution-of-the-solar-system-with-geometric-numerical-integrators

Take a look at figures 5 and 6 (again!):

This is what is being modelled:

And look at the phase diagrams:


It's from your own reference! Need I say more?

That sounds more like something you should educate yourself about. Harmonic motion and circular motion are equivalent. See this two minute video:



This is why it says the 'trajectory of the exact solution' in the caption. Further down in the document it says: "This shows that the numerical error in the energy always remains small with size h. Indeed,we observe in Figure 6 (right picture) that the numerical trajectory (in blue)remains close to the exact one (in red)."

It says 'trajectory' in reference to a diagram of a circular path because the example is harmonic oscillation transformed into circular motion. Circular motion can be represented as harmonic motion and vice versa.

If the spring is losing energy and slowing down, the circular motion would be seen to spiral down, and if it was gaining energy it would spiral out. The 'symplectic' version where the energy is maintained would be perpetual motion in a spring device.

From the other document we see the same examples of circular motion without reference to springs or harmonic oscillators:

http://astro.pas.rochester.edu/~aquillen/phy411/lecture7.pdf

It seems to me that most FEers reject Astronomy, Astrophysics, cosmology and the related sciences. 

This is rich considering the number of your co-religionists here unable to do more than recite scripture.