[Vo]:does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 2011.02.18

[Rich Murray]

does classical mechanics always fail to predict or retrodict for 3 or
more Newtonian gravity bodies? Rich Murray 2011.02.18

Hello Steven V Johnson,

Can I have a free copy of the celestial mechanics software to run on
my Vista 64 bit PC?

In fall, 1982, I wrote a 200-line program in Basic for the
Timex-Sinclair $100 computer with 20KB RAM that would do up to 4
bodies in 3D space or 5 in 2D space, about 1000 steps in an hour,
saving every 10th position and velocity -- I could set it up to
reverse the velocities after the orbits became chaotic after 3 1/2
orbits from initial perfect symmetry as circles about the common
center of gravity, finding that they always maintained chaos, never
returning to the original setup -- doubling the number of steps while
reducing the time interval by half never slowed the the evolution of
chaos by 3 1/2 orbits -- so I doubted that there is any mathematical
basis for the claim that classical mechanics predicts the past or
future evolution of any system with over 2 bodies, leading to a
conjecture that no successful algorithm exists, even without any close
encounters.

Has this been noticed by others?

Rich Murray rmfor...@gmail.com  505-819-7388
1943 Otowi Road, Santa Fe, New Mexico 87505

On Fri, Feb 18, 2011 at 4:30 PM,
OrionWorks - "Steven V Johnson"  wrote:

> Just a brief side-comment...
>
> Some of this "lingo" is fascinating stuff to me. Having performed a
> lot of theoretical computer simulation work on my own using good'ol
> fashion Newtonian based Celestial Mechanics algorithms, where
> typically I use "a = 1/r^2", I noticed orbital pattern behavior
> transforms into something RADICALLY different, such as if I were to
> change the classical algorithm to something like "a = 1/r^3". You can
> also combine both of them like "a = 1/r^2 +/-  1/r^3" within the same
> computer algorithm. That produces interesting side effects too. I'm
> still trying to get a handle on it all.
>
> Regards
> Steven Vincent Johnson
> www.OrionWorks.com
> www.zazzle.com/orionworks

RE: [Vo]:does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 2011.02.18

[OrionWorks - Steven Vincent Johnson]

Hi Rick,

I've been very busy with all the rabble rousing going on at the State
Capital. I'm currently uploading more videos of the situation at the
Capital.

Don't know if I can answer your question thoroughly. But I'll do what I can.

I'm not an expert on the matter.  However it's my understanding that when
you introduce at least three bodies it is impossible to predict what will
happen to orbital bodies. From my own experience it can also be difficult to
predict what will happen with just two bodies mucking about, depending on
the original vectors fed into the algorithm.

The Celestial Mechanics "s/w" I've been working on is written in VisualBasic
.NET. It's a forgiving language, and I'm sick and tired of dealing with C++.
My CM programs still under development. I'm still on windows'XP, 32 bit.

Congratulations on your Sinclair project. I started on a TRS-80.

Regards,
Steven Vincent Johnson
www.OrionWorks.com
www.zazzle.com/orionworks

Re: [Vo]:does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 2011.02.18

[Charles Hope]

I'm thinking your findings of irreversibility reflected the idiosyncrasies of 
floating point math represented in binary numbers, and not the physics itself. 

Sent from my iPhone. 

On Feb 18, 2011, at 22:17, Rich Murray  wrote:

> does classical mechanics always fail to predict or retrodict for 3 or
> more Newtonian gravity bodies? Rich Murray 2011.02.18
> 
> Hello Steven V Johnson,
> 
> Can I have a free copy of the celestial mechanics software to run on
> my Vista 64 bit PC?
> 
> In fall, 1982, I wrote a 200-line program in Basic for the
> Timex-Sinclair $100 computer with 20KB RAM that would do up to 4
> bodies in 3D space or 5 in 2D space, about 1000 steps in an hour,
> saving every 10th position and velocity -- I could set it up to
> reverse the velocities after the orbits became chaotic after 3 1/2
> orbits from initial perfect symmetry as circles about the common
> center of gravity, finding that they always maintained chaos, never
> returning to the original setup -- doubling the number of steps while
> reducing the time interval by half never slowed the the evolution of
> chaos by 3 1/2 orbits -- so I doubted that there is any mathematical
> basis for the claim that classical mechanics predicts the past or
> future evolution of any system with over 2 bodies, leading to a
> conjecture that no successful algorithm exists, even without any close
> encounters.
> 
> Has this been noticed by others?
> 
> Rich Murray rmfor...@gmail.com  505-819-7388
> 1943 Otowi Road, Santa Fe, New Mexico 87505
> 
> On Fri, Feb 18, 2011 at 4:30 PM,
> OrionWorks - "Steven V Johnson"  wrote:
> 
>> Just a brief side-comment...
>> 
>> Some of this "lingo" is fascinating stuff to me. Having performed a
>> lot of theoretical computer simulation work on my own using good'ol
>> fashion Newtonian based Celestial Mechanics algorithms, where
>> typically I use "a = 1/r^2", I noticed orbital pattern behavior
>> transforms into something RADICALLY different, such as if I were to
>> change the classical algorithm to something like "a = 1/r^3". You can
>> also combine both of them like "a = 1/r^2 +/-  1/r^3" within the same
>> computer algorithm. That produces interesting side effects too. I'm
>> still trying to get a handle on it all.
>> 
>> Regards
>> Steven Vincent Johnson
>> www.OrionWorks.com
>> www.zazzle.com/orionworks
> 

RE: [Vo]:does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 2011.02.18

[OrionWorks - Steven Vincent Johnson]

> I'm thinking your findings of irreversibility reflected the idiosyncrasies
of
> floating point math represented in binary numbers, and not the physics
itself.

I'm not sure what you mean by "irreversibility" but if you are referring to
my Celestial Mechanics computer programs, I have never stated that what I'm
doing reflects "physics itself"

I am interested in the mathematical aspects of these algorithms that are
often used to explain "physics" primarily from a heuristic perspective.

I'm also interested in the mathematical characteristics of chaos that can
also be introduced using the same algorithms.

I have many interests and objectives. "Physics" is simply a subset of it.

* * * * *

PS: In the meantime please feel free to check out my latest batch of You
Tube videos of the continued pandemonium (called democracy in action)
occurring on the square and within the state capital of Madison, Wisconsin,
concerning protests against Governor Scott Walker's bill that would destroy
50 years of the right to bargain collectively.

For the record: Union leaders have basically told the governor that we would
be willing to agree to pretty much all harsh the fiscal constraints the
Scott Walker listed in his bill if he would remove the part that destroys
the right to bargain collectively. The governor said no. He refuses to back
down from his original bill.

I think that pretty much tells us what Scott Walker's real objective is. If
it was just about the money, as he has all along claimed, then the problem
would be solved. Apparently it isn't. He wants to destroy the unions so that
he never has to deal with them again... EVER. He has only been in office for
two months. What a way to start out winning friends and influencing people.
. He attempted to ram his version of the "bill" down our throats in a matter
of six days with very little notice and NO DISCUSSION. We were almost
blitzed. This is why we are so upset and won't back down. This is what
democracy is all about when someone attempts to disrespect what democracy is
all about.

http://www.youtube.com/user/OrionworksVideos


Regards
Steven Vincent Johnson
www.OrionWorks.com
www.zazzle.com/orionworks

Re: [Vo]:does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 2011.02.18

[Rich Murray]

The only access to "the physics itself" we have with finite nervous
systems is by using digital approximations with finite number strings,
processed by algorithms of finite instruction size, so there are
always round-off errors, which always diverge without limit, even if
there are no close encounters.  So, it's a huge leap of faith to
assume that the "present data" for a certain finite time interval
actually allows prediction of a single future path or retrodiction of
a single past path -- ie, classical mechanics probably can be proved
to be incurably flawed, while allowing a certain amount of qualified
estimation of probable paths forward and backward in time for the
first 3 "orbits" or so...

I've read that actually the 3-body problem does have exact general
solutions, which involve such long, very slowly converging sequences
of terms, as to be practically unworkable in practice.  Probaby, it
can be shown that the energy needed to run an ideal finite digital
computer until a certain limit of accuracy is reached (testable by
running the same problem in parallel with identical computers,
watching to see at what point the results start to scatter) will grow
so fast with time and accuracy as to exhaust the energy available in
any universe that supports the computer...

Probably someone has already studied this...

It's not just that shit happens -- "happens" happens...

So, in reality, the "present" interval, however brief in time and tiny
in space, necessarily in complex interaction with a possibly infinite
external universe or hyperverse, must be inexplicable, "causeless",
ie, totally "magical"...

This has in recent thousands of years been a common insight for
advanced explorers of expanded awareness in many traditions.

Rich Murray "lookslikeallthoughtiswrong"@godmail.com


On Fri, Feb 18, 2011 at 9:50 PM, Charles Hope
 wrote:
> I'm thinking your findings of irreversibility reflected the idiosyncrasies of 
> floating point math represented in binary numbers, and not the physics itself.
>
> Sent from my iPhone.
>
> On Feb 18, 2011, at 22:17, Rich Murray  wrote:
>
>> does classical mechanics always fail to predict or retrodict for 3 or
>> more Newtonian gravity bodies? Rich Murray 2011.02.18
>>
>> Hello Steven V Johnson,
>>
>> Can I have a free copy of the celestial mechanics software to run on
>> my Vista 64 bit PC?
>>
>> In fall, 1982, I wrote a 200-line program in Basic for the
>> Timex-Sinclair $100 computer with 20KB RAM that would do up to 4
>> bodies in 3D space or 5 in 2D space, about 1000 steps in an hour,
>> saving every 10th position and velocity -- I could set it up to
>> reverse the velocities after the orbits became chaotic after 3 1/2
>> orbits from initial perfect symmetry as circles about the common
>> center of gravity, finding that they always maintained chaos, never
>> returning to the original setup -- doubling the number of steps while
>> reducing the time interval by half never slowed the the evolution of
>> chaos by 3 1/2 orbits -- so I doubted that there is any mathematical
>> basis for the claim that classical mechanics predicts the past or
>> future evolution of any system with over 2 bodies, leading to a
>> conjecture that no successful algorithm exists, even without any close
>> encounters.
>>
>> Has this been noticed by others?
>>
>> Rich Murray rmfor...@gmail.com  505-819-7388
>> 1943 Otowi Road, Santa Fe, New Mexico 87505
>>
>> On Fri, Feb 18, 2011 at 4:30 PM,
>> OrionWorks - "Steven V Johnson"  wrote:
>>
>>> Just a brief side-comment...
>>>
>>> Some of this "lingo" is fascinating stuff to me. Having performed a
>>> lot of theoretical computer simulation work on my own using good'ol
>>> fashion Newtonian based Celestial Mechanics algorithms, where
>>> typically I use "a = 1/r^2", I noticed orbital pattern behavior
>>> transforms into something RADICALLY different, such as if I were to
>>> change the classical algorithm to something like "a = 1/r^3". You can
>>> also combine both of them like "a = 1/r^2 +/-  1/r^3" within the same
>>> computer algorithm. That produces interesting side effects too. I'm
>>> still trying to get a handle on it all.
>>>
>>> Regards
>>> Steven Vincent Johnson
>>> www.OrionWorks.com
>>> www.zazzle.com/orionworks
>>
>
>

Re: [Vo]:does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 2011.02.18

[Charles Hope]

Yes, the Devil is in the details. It pays to know just how much Devil is in 
there, and in old school 8 bit BASIC, there is much. 

Classical Mechanics gives results that are reversible. So if the model isn't, 
it's just a numerical flaw, and not a profound fact about physics.  



Sent from my iPhone. 

On Feb 19, 2011, at 1:57, Rich Murray  wrote:

> The only access to "the physics itself" we have with finite nervous
> systems is by using digital approximations with finite number strings,
> processed by algorithms of finite instruction size, so there are
> always round-off errors, which always diverge without limit, even if
> there are no close encounters.  So, it's a huge leap of faith to
> assume that the "present data" for a certain finite time interval
> actually allows prediction of a single future path or retrodiction of
> a single past path -- ie, classical mechanics probably can be proved
> to be incurably flawed, while allowing a certain amount of qualified
> estimation of probable paths forward and backward in time for the
> first 3 "orbits" or so...
> 
> I've read that actually the 3-body problem does have exact general
> solutions, which involve such long, very slowly converging sequences
> of terms, as to be practically unworkable in practice.  Probaby, it
> can be shown that the energy needed to run an ideal finite digital
> computer until a certain limit of accuracy is reached (testable by
> running the same problem in parallel with identical computers,
> watching to see at what point the results start to scatter) will grow
> so fast with time and accuracy as to exhaust the energy available in
> any universe that supports the computer...
> 
> Probably someone has already studied this...
> 
> It's not just that shit happens -- "happens" happens...
> 
> So, in reality, the "present" interval, however brief in time and tiny
> in space, necessarily in complex interaction with a possibly infinite
> external universe or hyperverse, must be inexplicable, "causeless",
> ie, totally "magical"...
> 
> This has in recent thousands of years been a common insight for
> advanced explorers of expanded awareness in many traditions.
> 
> Rich Murray "lookslikeallthoughtiswrong"@godmail.com
> 
> 
> On Fri, Feb 18, 2011 at 9:50 PM, Charles Hope
>  wrote:
>> I'm thinking your findings of irreversibility reflected the idiosyncrasies 
>> of floating point math represented in binary numbers, and not the physics 
>> itself.
>> 
>> Sent from my iPhone.
>> 
>> On Feb 18, 2011, at 22:17, Rich Murray  wrote:
>> 
>>> does classical mechanics always fail to predict or retrodict for 3 or
>>> more Newtonian gravity bodies? Rich Murray 2011.02.18
>>> 
>>> Hello Steven V Johnson,
>>> 
>>> Can I have a free copy of the celestial mechanics software to run on
>>> my Vista 64 bit PC?
>>> 
>>> In fall, 1982, I wrote a 200-line program in Basic for the
>>> Timex-Sinclair $100 computer with 20KB RAM that would do up to 4
>>> bodies in 3D space or 5 in 2D space, about 1000 steps in an hour,
>>> saving every 10th position and velocity -- I could set it up to
>>> reverse the velocities after the orbits became chaotic after 3 1/2
>>> orbits from initial perfect symmetry as circles about the common
>>> center of gravity, finding that they always maintained chaos, never
>>> returning to the original setup -- doubling the number of steps while
>>> reducing the time interval by half never slowed the the evolution of
>>> chaos by 3 1/2 orbits -- so I doubted that there is any mathematical
>>> basis for the claim that classical mechanics predicts the past or
>>> future evolution of any system with over 2 bodies, leading to a
>>> conjecture that no successful algorithm exists, even without any close
>>> encounters.
>>> 
>>> Has this been noticed by others?
>>> 
>>> Rich Murray rmfor...@gmail.com  505-819-7388
>>> 1943 Otowi Road, Santa Fe, New Mexico 87505
>>> 
>>> On Fri, Feb 18, 2011 at 4:30 PM,
>>> OrionWorks - "Steven V Johnson"  wrote:
>>> 
 Just a brief side-comment...
 
 Some of this "lingo" is fascinating stuff to me. Having performed a
 lot of theoretical computer simulation work on my own using good'ol
 fashion Newtonian based Celestial Mechanics algorithms, where
 typically I use "a = 1/r^2", I noticed orbital pattern behavior
 transforms into something RADICALLY different, such as if I were to
 change the classical algorithm to something like "a = 1/r^3". You can
 also combine both of them like "a = 1/r^2 +/-  1/r^3" within the same
 computer algorithm. That produces interesting side effects too. I'm
 still trying to get a handle on it all.
 
 Regards
 Steven Vincent Johnson
 www.OrionWorks.com
 www.zazzle.com/orionworks
>>> 
>> 
>> 
> 

Re: [Vo]:does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 2011.02.18

[Stephen A. Lawrence]

On 02/18/2011 10:17 PM, Rich Murray wrote:
> does classical mechanics always fail to predict or retrodict for 3 or
> more Newtonian gravity bodies? Rich Murray 2011.02.18
> [ ... ]
>   
> In fall, 1982, I wrote a 200-line program in Basic for the
> Timex-Sinclair $100 computer with 20KB RAM that would do up to 4
> bodies in 3D space... 
> [ ... ]
> so I doubted that there is any mathematical
> basis for the claim that classical mechanics predicts the past or
> future evolution of any system with over 2 bodies, leading to a
> conjecture that no successful algorithm exists, even without any close
> encounters.
>
> Has this been noticed by others?

See, for example,

http://en.wikipedia.org/wiki/Stability_of_the_Solar_System#Digital_Orrery


There are also far better algorithms than what you were using, which,
I'm sure, was a simple integrator of the nonlinear system of equations. 
Simply cutting the time step doesn't do much for you if the basic
algorithm isn't very accurate.

See, for example,

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ5-46DFTHW-8W&_user=10&_coverDate=12%2F31%2F1987&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C50221&_version=1&_urlVersion=0&_userid=10&md5=59646ea61335b206d3a7cea0bed0ce8d&searchtype=a


(sorry, I don't have the full text, but the abstract sounds interesting.)

Re: [Vo]:does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 2011.02.18

[Rich Murray]

Hello Stephen A. Lawrence,

Thanks for the informative answer.  It'd be impressive if the most
accurate methods since this review in 1987 agree with each other far
into the future and past -- how can we find out the details about
results for the 3-body problem, in commonsense terms?  Is this
accessible for PC users?  Could a business sell the program and run a
collaborative blog for users?

Laskar #1

In 1989, Jacques Laskar of the Bureau des Longitudes in Paris
published the results of his numerical integration of the Solar System
over 200 million years. These were not the full equations of motion,
but rather averaged equations along the lines of those used by
Laplace. Laskar's work showed that the Earth's orbit (as well as the
orbits of all the inner planets) is chaotic and that an error as small
as 15 metres in measuring the position of the Earth today would make
it impossible to predict where the Earth would be in its orbit in just
over 100 million years' time.

[edit]Laskar & Gastineau

Jacques Laskar and his colleague Mickaël Gastineau in 2009 took a more
thorough approach by directly simulating 2500 possible futures.
Each of the 2500 cases has slightly different initial conditions:
Mercury's position varies by about 1 metre between one simulation and
the next.[13]

In 20 cases, Mercury goes into a dangerous orbit and often ends up
colliding with Venus or plunging into the sun.
Moving in such a warped orbit, Mercury's gravity is more likely to
shake other planets out of their settled paths:
in one simulated case its perturbations send Mars heading towards Earth.[14]

13. ^ "Solar system's planets could spin out of control".
newscientist. Retrieved 2009-06-11.

14. ^ "Existence of collisional trajectories of Mercury, Mars and
Venus with the Earth". Retrieved 2009-06-11.

http://www.nature.com/nature/journal/v459/n7248/full/nature08096.html

Letter
Nature 459, 817-819 (11 June 2009)
doi:10.1038/nature08096; Received 17 February 2009; Accepted 22 April 2009

ARTICLE LINKS
Figures and tables
Supplementary info
SEE ALSO
News and Views by Laughlin
Editor's Summary

Existence of collisional trajectories of Mercury, Mars and Venus with the Earth

J. Laskar 1 & M. Gastineau 1

Astronomie et Systèmes Dynamiques, IMCCE-CNRS UMR8028, Observatoire de
Paris, UPMC, 77 Avenue Denfert-Rochereau, 75014 Paris, France
Correspondence to: J. Laskar 1 Correspondence and requests for
materials should be addressed to J.L. (Email: las...@imcce.fr ).

Abstract

It has been established that, owing to the proximity of a resonance
with Jupiter, Mercury’s eccentricity can be pumped to values large
enough to allow collision with Venus within 5 Gyr (refs 1–3).
This conclusion, however, was established either with averaged
equations 1, 2 that are not appropriate near the collisions or with
non-relativistic models in which the resonance effect is greatly
enhanced by a decrease of the perihelion velocity of Mercury 2, 3. In
these previous studies, the Earth’s orbit was essentially unaffected.
Here we report numerical simulations of the evolution of the Solar
System over 5 Gyr, including contributions from the Moon and general
relativity.
In a set of 2,501 orbits with initial conditions that are in agreement
with our present knowledge of the parameters of the Solar System, we
found, as in previous studies 2,
that one per cent of the solutions lead to a large increase in
Mercury’s eccentricity -- an increase large enough to allow collisions
with Venus or the Sun.
More surprisingly, in one of these high-eccentricity solutions, a
subsequent decrease in Mercury’s eccentricity induces a transfer of
angular momentum from the giant planets that destabilizes all the
terrestrial planets ~3.34 Gyr from now, with possible collisions of
Mercury, Mars or Venus with the Earth.

Astronomie et Systèmes Dynamiques, IMCCE-CNRS UMR8028, Observatoire de
Paris, UPMC, 77 Avenue Denfert-Rochereau, 75014 Paris, France
Correspondence to: J. Laskar 1 Correspondence and requests for
materials should be addressed to J.L. (Email: las...@imcce.fr ).

So, with the most accurate methods, 1% of <5x10^9 Earth orbits lead to
chaos -- but also occurring in the solar system in that time are
changes via civilizations, solar evolution, major meteor impacts,
intra solar system gas density and temperature changes, about 20
orbits around the Galactic center, with resulting encounters with dark
matter flows and the Galactic plane, and things that go bump in the
night...

Rich

On Sat, Feb 19, 2011 at 1:59 PM, Stephen A. Lawrence  wrote:
>
>
> On 02/18/2011 10:17 PM, Rich Murray wrote:
>
> does classical mechanics always fail to predict or retrodict for 3 or
> more Newtonian gravity bodies? Rich Murray 2011.02.18
> [ ... ]
>
>
> In fall, 1982, I wrote a 200-line program in Basic for the
> Timex-Sinclair $100 computer with 20KB RAM that would do up to 4
> bodies in 3D space...
> [ ... ]
> so I doubted that there is any mathematical
> basis for the claim that classical mechanics pr

RE: [Vo]:does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 2011.02.18

[Abd ul-Rahman Lomax]

At 11:12 PM 2/18/2011, OrionWorks - Steven Vincent Johnson wrote:

Congratulations on your Sinclair project. I started on a TRS-80.


Heh! Well, *I* -- the word is drawn out -- started on an Altair 8800. 
Pthtpthhh! 

RE: [Vo]:does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 2011.02.18

[OrionWorks - Steven Vincent Johnson]

> Jacques Laskar and his colleague Mickal Gastineau in 2009 took a more
> thorough approach by directly simulating 2500 possible futures.
> Each of the 2500 cases has slightly different initial conditions:
> Mercury's position varies by about 1 metre between one simulation and
> the next.[13]
> 
> In 20 cases, Mercury goes into a dangerous orbit and often ends up
> colliding with Venus or plunging into the sun.
> Moving in such a warped orbit, Mercury's gravity is more likely to
> shake other planets out of their settled paths:
> in one simulated case its perturbations send Mars heading towards Earth.[14]


Based on my own heuristic studies of orbital characteristics it's very easy for 
me to speculate that all the planets in our solar system are in chaotic orbits, 
particularly if you factor in a sufficient amount of geological time into the 
equation. IOW, it's a very slow process, for the most part. Nevertheless, chaos 
may be the norm, not the exception. Seems to me that chaos could also explain a 
lot of dramatic climatic changes our planet has experienced since it first 
formed 4.5 billion years ago.

Just another wobble, give or take several million years.

Regards,
Steven Vincent Johnson
www.OrionWorks.com
www.zazzle.com/orionworks

Re: [Vo]:does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 2011.02.18

[Abd ul-Rahman Lomax]

At 10:17 PM 2/18/2011, Rich Murray wrote:

does classical mechanics always fail to predict or retrodict for 3 or
more Newtonian gravity bodies? Rich Murray 2011.02.18


I think there is a misconception here. There isn't any true two-body 
or three-body problem because there are far, far more than two or 
three bodies in the universe!


We simplify problems by neglecting what is remote. So we might, 
indeed, look at 3-body problems; some solutions are known that are 
special cases, if I'm correct. As the attempt to predict extends into 
the future, however, the results become more and more inaccurate, 
except in stable special cases.


I don't recall description of the overall problem mentioned when I 
was young, before chaos theory became well-known. The problem is 
infinite sensitivity to initial conditions. In setting up an attempt 
to predict behavior of a system, even when the laws of motion are 
well-defined, it's necessary to specify the initial conditions, i.e., 
the position and velocity of the elements. Now, from the Uncertainty 
Principle, we can only know these to a certain combined accuracy, the 
product of the uncertainties cannot be less than a fixed value.


But surely that's only a tiny detail!

However, turns out, some physical systems are infinitely sensitive to 
initial conditions. Real physical systems, some fairly simple ones. 
Using math, start with one particular exact initial condition, and 
you get one result. Start from something infinitesimally different, 
you can get a radically different result.


In practice, this means that the future of a system cannot, in 
general, be exactly predicted, and for long periods of time, 
relatively, the inaccuracy can become gross. There is a lovely 
youtube video showing a pendulum suspended over four magnets. If you 
start from a particular starting position, hovering over which magnet 
will the pendulum end up settling? Outside regions close to the 
magnets, it turns out to be *unpredictable.* That's because one 
cannot set the initial conditions *exactly* the same. You can't 
predict the outcome even by a history of tries, by releasing the 
pendulum again from the supposedly same spot. You can't make the spot 
'same' enough. (Probably. There might exist some regions where the 
outcome is predictable, besides the obvious ones over the attractors.)


http://www.youtube.com/watch?v=Qe5Enm96MFQ&feature=related