three-body problem

"radically different ... solutions"? Doesn't that make you suspicious/nervous? ...Jim Thompson

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| James E.Thompson                                 |    mens     | 
| Analog Innovations                               |     et      | 
| Analog/Mixed-Signal ASIC's and Discrete Systems  |    manus    | 
| STV, Queen Creek, AZ 85142    Skype: skypeanalog |             | 
| Voice:(480)460-2350  Fax: Available upon request |  Brass Rat  | 
| E-mail Icon at http://www.analog-innovations.com |    1962     | 

             I'm looking for work... see my website. 

Thinking outside the box...producing elegant & economic solutions.

Of course it's confounded mathematicians, they were looking for closed-form solutions but the overwhelming majority of real-world physics problems don't have them.

You can't write an exact closed-form equation of how a nuclear weapon works either called "The Nuclear Bomb Equation" that gives an explicit answer in elementary functions for fireball diameter from first principles either. They seem to work OK though

Amazing but plz take a course in scientific computing/numerical methods prior to concluding that everything is a lie based on friggin' LTSpice behaving weird because you changed the timestep.

Also even the general three-body problem isn't universally unboundedly chaotic; for any given phase-space volume it will have regions where the Hamiltonian is more sensitive to initial conditions and small perturbations from equilibrium, and others less so.

No, I just fiddle until I see the solution that I want.

--

John Larkin         Highland Technology, Inc 
picosecond timing   precision measurement  

jlarkin att highlandtechnology dott com 
http://www.highlandtechnology.com

Numerical solutions to the simplified 3-body probelm are just as intractable. The benign cases aren't very interesting.

Did I say that everything is a lie? I don't remember doing that.

--

John Larkin         Highland Technology, Inc 
picosecond timing   precision measurement  

jlarkin att highlandtechnology dott com 
http://www.highlandtechnology.com

The whole world should make him nervous, then, because 100% deterministic physical systems only exist on the blackboards of mathematicians.

And if LTSpice is giving radically different outcomes due to small changes in circuit values or timestep my money is that the circuit design itself has chaotic properties, not that there's anything intrinsically wrong with LTSpice. Or bumping up against roundoff/truncation error.

Nonsense, for any _specific_ set of initial conditions the three body problem is completely deterministic. You can write a computer program consisting of three idealized masses which are perfect spheres in empty space under the influence of gravity, set it up with some initial conditions, and the computer will predict the motion of three idealized masses which are perfect spheres in empty space under the influence of gravity perfectly down to the implied resolution of your numerical precision.

What other kind of "numerical solution" are you looking for?

Oh! I see! Optimization >:-} ...Jim Thompson

--
| James E.Thompson                                 |    mens     | 
| Analog Innovations                               |     et      | 
| Analog/Mixed-Signal ASIC's and Discrete Systems  |    manus    | 
| STV, Queen Creek, AZ 85142    Skype: skypeanalog |             | 
| Voice:(480)460-2350  Fax: Available upon request |  Brass Rat  | 
| E-mail Icon at http://www.analog-innovations.com |    1962     | 

             I'm looking for work... see my website. 

Thinking outside the box...producing elegant & economic solutions.

Actually, in a quantum world, it's not. Or in a real world, with real objects in a real universe.

You can write a computer program

Change the mass of one of those ideal spheres by 1 LSB of your float, sim a while, and one of those expensive perfect spheres might get flung out of the system. Or not.

Before IEEE floats, different computers would round slightly differently, or compute transcendentals a tiny bit differently, so nonlinear sims like weather would produce wildly different results if run on different machines. Now, everyone has agreed on a uniformly incorrect result.

As an engineer, I care if a numerical solution is predictive of the future state of the system to some useful sort of accuracy... like maybe, let's get the sign right. Some systems can be simulated to parts per million over any time span. Chaotic systems can't.

--

John Larkin         Highland Technology, Inc 
picosecond timing   precision measurement  

jlarkin att highlandtechnology dott com 
http://www.highlandtechnology.com

What, me nervous? The whole world is amusing, especially the bizarre and unpredictable bits.

LT Spice does strange things sometimes with circuits that are obviously stable in real life. It's prudent to futz with things until you get a solution that makes sense. One trick is to play with the time step and plunk it somewhere inside a zone where nothing changes.

--

John Larkin         Highland Technology, Inc 
picosecond timing   precision measurement  

jlarkin att highlandtechnology dott com 
http://www.highlandtechnology.com

Right, expecting idealized mathematical models to translate directly to the real world doesn't make much sense. But for all intents and purposes the real world at the human-ish scale is deterministic. There is uncertainty, but that's only because we don't know the outcome yet and it's in the future. Whether we have "real" free will or just "free will" is an interesting question that comes out of this which is probably better expatiated on by others

That's true, but so what? The unconstrained perfect-three-body-in-an-empty-universe-problem doesn't exist in nature. Any system so wildly chaotic would've flung itself apart ages ago, or settled into at least some kind of metastable state. If it's in some kind of limit cycle or metastable state then that's a whole new ball game when it comes to analysis.

Weather predictions seem overall pretty bang-on where I am, so long as you aren't looking much more than a week in the future. And New England isn't really known for having predictable weather.

It doesn't make sense to think of a clear-cut line of division between "chaotic" systems and "non-chaotic" systems. There is no such thing. There are strongly deterministic systems, strongly chaotic systems, and everything in-between.

Rather, you can create that division, but only on a mathematician's blackboard. Not in the real world.

Not quite true; the three-body problem has a number of solutions, and it confounds PHYSICISTS, who deal with the real world, not mathematicians (who don't care how inaccurate the real world is, the equations are fine!).

It's an interesting quantum mechanics homework problem, though, to bounce one perfect spherical ball atop another, held stationary. Using Heisenberg uncertainty limits, preduct the number of bounces before the top ball is so far offcenter that it doesn't hit the bottom one!

Similarly, predicting the phase of a Wein bridge oscillator after startup, or that of an organ pipe, or the direction a twirled top will roll... or a hurricane's path, or the timing of an avalanche... not really predictable aspects of the physics.

Frequency of the Wein bridge oscillator, on the other hand, and Geiger (avalanche) tube click rates, are measurement-grade aspects of the exact same systems.

One really interesting system is the superregenerative receiver, that continually teases the stability boundary. It's chaotic as heck. I made a 1-transistor superregen receiver when I was a kid, and it brought in Radio Moscow loud and clear. Superregens aren't popular any more... gain used to be expensive and now it's almost free.

There must be an optical equivalent. I'm guessing this could be demonstrated with a semiconductor laser pretty easily.

--

John Larkin         Highland Technology, Inc 

lunatic fringe electronics

This is really misleading. The idea that there is not an analytical solution for any problem, has no significance at all. We have computers. There is, essentially, no non trivial real world problem that has a closed form solution.

Its easy to get into the non-reality trap of think that there is some intrinsic "I am clever" merit in solving equations by manipulating symbols. It fact its something that usually hinders solving problems.

-- Kevin Aylward

- SuperSpice

Well. Galaxies have huge numbers of stars in their cores, following erratic paths. They get captured, ejected, and yes even collided and swallowed, from time to time. But the population certainly seems to maintain itself.

Out of the some millions of asteroids expected to be floating around in our solar system, more than a few have been found on erratic paths, e.g.

It's in such a metastable cycle, where it will remain for a few thousand years and then bounce into some other orbit (or be ejected or other, of course).

Er, well, chaos theory is a superset of deterministic theory. If it's even "slightly" non-deterministic, it is by definition chaotic. It's like saying: "e is exactly two point seven eight, one eight two eight, and so on": a rational is a teeny tiny point that can't possibly equate to a transcendental number. Or, taking these literally:

:-)

Tim

--
Seven Transistor Labs, LLC 
Electrical Engineering Consultation and Contract Design 
Website: http://seventransistorlabs.com

You obviously haven't tried doing that, or you'd know better. Multi-body mechanics is an exponential noise amplifier like weather.

If you has a lossless pool table, in about 30 seconds after a break, the Heisenberg uncertainty associated with the positions of the balls would grow to be bigger than their diameter--you can't even know in advance which ones will collide.

The universe really isn't deterministic.

Cheers

Phil Hobbs

Solving equations is a mechanical skill. Worfram annoyed a lot of people when he (and others) wrote programs to solve symbolic algebra and calculus equations. If playing with equations gives a person insight into what's really going on, that's great. If not - the usual case - you may as well let some phone app do that for you, along with the normal four functions. Or numerically simulate and skip the equations.

Of course ideas have to come from somewhere, and we need ways to test them to see if they are practical or stupid. We spent all last week developing a stupid idea, down to preliminary PCB parts placement. Now we need to back off and find a somewhat less stupid idea.

ps - self-protecting mosfets are way too tricky.

--

John Larkin         Highland Technology, Inc 

lunatic fringe electronics

Nope. A chaotic system is an exponential noise amplifier--it's the sensitivity to initial conditions that's the determining factor. The Heisenberg uncertainty in a two-body orbit makes it nondeterministic, strictly speaking, but there's no exponential amplification, so it isn't chaotic.

Billiard balls are exponential amplifiers because if the impact position is off by a hair, the rebound angle is off a bit. The angular error turns into a position error that grows as the ball rolls along, making the next collision off by more, so the rebound angle is off by still more, and so on. Doesn't take long for that to get completely out of hand. Multi-body orbits are exponential amplifiers for the same reason.

While the uncertainty in the object positions formally grows without bound, chaotic systems often have energy integrals. For instance, the Galaxy may eject stars sometimes, but it can't randomly come completely apart because there isn't enough energy in the system. Similarly, the billiard balls might not have enough energy for one of them to leave the table by bouncing over the rails, so the possible positions are limited to the area of the felt.

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs 
Principal Consultant 
ElectroOptical Innovations LLC 
Optics, Electro-optics, Photonics, Analog Electronics 

160 North State Road #203 
Briarcliff Manor NY 10510 

hobbs at electrooptical dot net 
http://electrooptical.net