The three-body
problem and new computational methods
"Chaos
can spoil the whole day"
The three body instability is a great mathematical event. We love to talk about the weather, but the science guy will not touch the three body instability because his career may become unpredictable. The weatherman can make a joke about the weather, and it may come with the territory, but the science guy cannot make light of his inability to calculate and tell us about the rules of the weather. You cannot ask for an equation describing the path of three bodies when gravity pulls them together.
The new scientific method: Discover and make new tractable systems
In a general three body system, there is an equation that gives us the force acting on each of the three interacting bodies. The force that pulls on any body is very well known, for it depends on one variable only -- it's the distance between bodies -- and so the net force on any body can be computed readily. However, there exists no path we could predict any of the bodies to take. In other words, there exists no general solution to the three force equations that would result in an ellipse, hyperbole, a corkscrew, or any some such shape that can be mathematically computed ahead of real-time and thus called tractable. Hence the name for this topic: The three-body problem.
Three or more gravitationally interacting bodies can and will travel along any number of infinite trajectories with infinite shapes and infinitely changing curvatures. There exists no formula able to foretell the one of the infinity of paths when, in general, no path is to be excluded from being the actual path. Moreover, the classical mechanics of Copernicus, Galileo, Newton, Kepler, and Einstein do not offer any mechanism that would diminish the occurrence of chaos. In general, three or more bodies subject to classical gravity will loop in and around, swirling forever in a nonrepeating dance of chaos and no "cosmological constant" will ever alter or evolve chaos to resemble the order we observe.
We do not know where three bodies (or N-bodies) are going
In the three body scenario, the computer calculates the path one point at a time, but taking discrete steps in time causes position errors to mount quickly and after a thousand or so steps the computer results are hopelessly irrelevant. Taking smaller time increments improves accuracy, but it will take more time to compute the path over a particular distance. Eventually, the machine will spend so much time computing, the problem becomes intractable while the error is getting close to zero but the error never becomes zero. We can and do call such systems chaotic, but that should not mask our frustration at not being able to predict the weather two days hence, while, in a more formal system where mathematical solutions exist, we can predict an eclipse to the minute ten years from now. So, Bunky, how did the universe get formalized? With a swift kick in the pants?
Our computing methods are outclassed by the way nature computes
The three body problem is intractable and in general unsolvable even if all data in the calculations are perfect, and even if we know the initial mass and distances exactly. This means that the guy who dreamed up the butterfly effect sold everybody short. Claiming that trimming of numbers due to computer's precision the tally at the end of the day is not exact is as preposterous as clipping pennies in transactions and then claiming that the sum of all fractions belongs in his pocket. The butterfly effect is but a red flying herring with trimmed wings for the purpose of explaining intractability with computer deficiency when the computer is really not the problem. Even if our calculations are infinitely exact, -- that is, perfect, the three body problem is still intractable.
Nature deals with chaos quite well and the three gravitationally interacting bodies travel on a path that is smooth, exact, and repeatable. But the best computer, even the ideal Turing machine on which the computer is based, can calculate the trajectory only approximately and only with ever-increasing error.
Butterfly effect -- go trim your own wings
Turing machine is at times called the universal machine, which is a label originated by Turing. Later on von Neumann also claimed it can solve any problem. The resources going into the universal computer are unlimited memory and unlimited time, and it is the unlimited time that is the tough ingredient. Nature, when it comes to gravitation, knows the force at every conceivable point in space with near-infinite exactness and without computational delay. Nature operates in real-time, not in need-more-time. Nature does not grant upon itself the luxury of unlimited time to determine the next position of a body.
It is for this very reason that any mathematical proof that is based on intractability is really invalid -- as, for example, Dedekind's "proof" equating irrational with real numbers. It can be said that any proof that happens 'eventually' is really not a proof.
It gets worse. The solar system has more than two bodies but we, including Newton and Kepler, are lucky to be able to treat any planet as a two-body sun-planet subsystem because the sun is much bigger than the rest of the planets. When we take a particular planet with the sun into consideration, the rest of the planets are small and so far away that these do not significantly enter into our calculations, and the orbit of each planet is mathematically solvable and therefore repeatable and therefore predictable. Asking the same question in a more appropriate way, Bunky, how did the solar system get lucky? How were the planets created as to form and maintain two-body systems?
"Universal" machine -- well, you cannot buy it with unlimited memory. But even if you could build it, it still will not do what nature does
The N-body problem has been kicked around mathematically since Newton. Recently, some news claim the 3-body and even N-body problems to be solved. This is a very tenuous and vulnerable scientific topic and one only needs to ask "Where is the equation?" since N-body problems do not have a general solution. The scientific vocabulary hates the word 'intractable,' but this is what happens when people attempt to emulate the N-body problem (with Turing machines) in competition with the way nature does it every second of every day, on a multi-galactic scale no less.
There are two issues here: First, the speed of the present day computing machine is not a defect or a cure because the N-body computational aspects are of different, quantum mechanical class. Secondly, the emulation of the N-body problem still does not address the self-organizing mechanism that is observed as numerous and wondrous geometries of cosmic assemblies.
When working QM into a machine, the implication of the first issue is that you do not want to emulate algebraic functions with QM. This is not because it is fashionable to say to think "outside the box," but because algebra cannot transcend intractability via another (QM) implementation. The second is that you will need to get comfortable with infinities -- the most difficult though rewarding task. There is something about the human brain that can do that and also 'that something' may be superior to other brains that are "out there." As far as the author is concerned, I say, "I want the infinity of the incommensurables, and I want them now!" Yes, this is about the infinite superposition and instant action of QM.
Done deal?
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QUANTUM
PYTHAGOREANS |
As convenient as it may be to declare things real and measurable and repeatable before these can be "scientific," the bottom line on this is that it will not get us any further than we are already. Really, the difference between a bicycle and a car is small and it is mostly about engineering rather than science. If anything, the ongoing ignorance about light and gravitation calls for the reevaluation of the scientific method for about eighty years now. As soon as QM benefits start to show up, funding for classical approaches will fall off regardless of one's convictions.
The current scientific method deals with measurability and repeatability. The superposition phenomena in QM simply does not let us keep track of everything every second of every day if we want to avail ourselves to the concurrency benefits of QM. In QM, there is no bound on concurrency but if you insist on tracking or measuring things then the concurrency cannot become unbounded.
The imposition of repeatability is fine for engineering as we appreciate our car powering up nicely every morning. QM setup, however, can produce different results, some better than others. Some results may be similar and some dissimilar, yet none would be wrong if there is nothing "wrong" with the setup. The results may not be repeatable at will and the conditions for repeatability can be complex or periodic. Yet, because of the infinite superposition that yields tractability, the results will happen in a practical timeframe. In QM, the repeatability can also be had within some range that is practical. All desired results will, moreover, yield solutions along a tractable time -- that is, the result may look like a miracle but once the result is repeatable, it will become a genuine solution. QM is about solutions to complex problems (technically non-polynomial problems) and there exist many unknown or hidden solutions. For example, the topology and dynamics of a spherical galaxy represents one counter-intuitive N-body solution. It is likely there are solutions that are not freely observed in nature or that do not yet exist in nature.
The new scientific method, then, is as follows: Discover and make new tractable systems. The discovery of new tractable systems is about making potentially chaotic systems into organized systems. New tractable systems represent a solution to multi-body and multi-wave interactions.
The point is that you cannot use tractable methods to arrive at new systems. Tractable methods are today's methods of at-will measurement but these discover "new" systems that are merely a subset of existing tractable systems. In fact, the present "scientific" method may be good for defining engineering. So, while the new definition does not speak of prescribed methods, it opens up the use of quantum mechanical methods that apply infinite concurrency among wavefunctions. [Of course], you know that decoherence is about measurement and you know that decoherence spoils the QM environment.
The new scientific method is not offered here for the sake of trying something new and different but because the insistence on the old scientific method will, scientifically, preclude us from new discoveries. Tractability is the solution and we find planets in their orbits tractable even though the gravitational interactions in general could be chaotic and intractable.
There are intractable methods that do not result in new systems. Represented by the traveling salesman problem, intractable methods are not the same as the infinite concurrency methods. Scientists that offer intractable solutions just point to dead ends.
Perhaps the best example of the inadequacy of the present method is the demand on observing neutrons during certain atomic core interactions. The insistence on the measurement of neutrons as "scientific proof" of reactions bounds the neutron and precludes QM from making its mark. It's time to "get out of the kitchen if you cannot take normal temperature."
The new scientific method: Discover and make new tractable systems
While we know that gravitation among three or more bodies is in general intractable, we know there exist tractable and practical solutions. Our own planet along with the other planets are a wonderful example of that. Similarly, a squaring of a circle is again, in general, insolvable, but what if there are specific solutions.
Many discussions are about when and how science "recognizes" psi research as a legitimate component of the scientific pursuit.
Any science that emphasizes hard measurability (that is, reality at all times) and repeatability will forever remain a component of engineering.
Any psience that does not deliver real goods will forever remain a component of the virtual, crunching data in the what-else domain which, inherently, lacks action.
In general, if-then methods are intractable when attempting to solve non-polynomial problems.
Real and Virtual are separated by a discontinuity and can be merged only at peril. Yet, transitioning between real and virtual is the only means of improving organization within individuals, groups, and machines.
Reflecting on recent events, the neutrino detector structures are a great feat of engineering that is built on science so irrelevant it borders on fraudulent.
Science, Psience, or What?
Many discussions boil down to 'intelligent design' vs. 'evolution.' Evolutionists like to say that the 'intelligent design' is really a form of a religion -- that is, belief. Not so. Big bang may at times be picked by religious groups to espouse the God's creation ideas. The proponents of intelligent design, however, do not think much of the big bang. Even though the big bang theory could be similar to, or viewed as, "creationist," there is no intelligence in big bang.
There is no intelligence in the 'strongest survive' mentality of Darwin, for every evolution theory that is tied to 'superior this or that' historically devolves into bare knuckles contests the likes of fascism, communism, and the original one-only system -- the Islam.
It can be said that intelligent design is the necessary ingredient in moving out of the zero-sum perception of the world. Zero-sum issues from left-brain dominance that is driven by the 'if-then' method. Right-brain thinking is driven by the 'what-else' method that creates new relevant contexts.
We have a book review 'Monk in The Garden' on Mendel's discoveries
There are answers to many questions and the HyperFlight site has many navigational posts to get you there. Your attitude, then, is to apply the obvious but the impossible notion that there is more than one way to the center of the maze. More ways you see, more confidence you will get in building on up from there. More ways you know, clearer your mind will be. More ways you feel, the separation of dependent and independent variables becomes more apparent as well.
What if, for example, the chaotic environment is an overlay of several possible solutions -- instead of representing an absence of a solution? Chaos workout may not be that difficult. Keep in mind the hyperbole, for you will need it each time you transition between the real and the virtual. See you there.
In the case of orbiting bodies, a three-body potentially chaotic system is averted when planets' orbits interlock in a ratio. A three-body system converts to a tractable two-body system. This is a common situation in Oss (our solar system) while multi-pointed star orbit shapes result.
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Added note on the
butterfly effect summer 2005
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