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May 15th, 2017, 07:01 AM   #1
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The N-Body problem, what can we know?

Is there a way to setup an n-body system and know what state it should be via "common sense" or a simple proof that works because the system is non-chaotic?

Let's say you have 3 bodies at the points of a equilateral triangle. They all pull on each other with the same force. Since the net force is basically a virtual particle that's the center of the other two, and all virtual particles are equilateral, the system should be predictable, non-chaotic. It's essentially 3 independent 2-body problems.
Are there other states we can prove at any given T? The above problem I think can be solved with integrals.
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May 15th, 2017, 05:03 PM   #2
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No. Chaos will emerge. I one of your other posts, Mashcke highlighted the idea of differential equations applied to a 'predator-prey' system. This is a good starting point.

In fact the Lotka-Volterra model helps us to understand chaos better due to its simplicity, maybe go have a look at those, then come back to the three body problem.

I mention this because in both cases we are dealing with a system of coupled differential equations. The three body problem involves 3 coupled second order differential equations. It has been shown on several accounts that chaos will be exhibited in these systems, and often is often said that dimensionality >2 is required.
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May 16th, 2017, 06:58 AM   #3
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Originally Posted by Joppy View Post
No. Chaos will emerge. I one of your other posts, Mashcke highlighted the idea of differential equations applied to a 'predator-prey' system. This is a good starting point.

In fact the Lotka-Volterra model helps us to understand chaos better due to its simplicity, maybe go have a look at those, then come back to the three body problem.

I mention this because in both cases we are dealing with a system of coupled differential equations. The three body problem involves 3 coupled second order differential equations. It has been shown on several accounts that chaos will be exhibited in these systems, and often is often said that dimensionality >2 is required.
Well, the idea is, if I can know an equilateral system with T, then I could create a program that brute forces the topology of logic that can come to those results. Then try it with a more complex system.
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May 16th, 2017, 08:03 PM   #4
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Originally Posted by InkSprite View Post
brute forces the topology of logic
?
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May 21st, 2017, 04:54 AM   #5
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Instead of talking. I'm going to focus on producing. There are a lot of programs I need to write.
I'm going to try to break down all aspects of an N-Body system.

Another repeating state is if you spin the equilateral points while they attract each other. if it rotates 60 degrees and the speeds are just right, you could possibly have a system that looks identical to its initial state.

Brute force pattern finding
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May 21st, 2017, 09:33 AM   #6
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Your program runs in discrete time, not continuous time. It's therefore an approximation. Because the system exhibits chaotic behaviour, the small deviations from reality that your model contains will throw the results off in unpredictable ways.

That's not to say that we can't predict what happens in certain trivial cases (when symmetry can be used to simplify the problem), but there's not a lot of interest in trivial cases (by definition).

Last edited by v8archie; May 21st, 2017 at 09:35 AM.
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May 21st, 2017, 01:48 PM   #7
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Your program runs in discrete time, not continuous time. It's therefore an approximation. Because the system exhibits chaotic behaviour, the small deviations from reality that your model contains will throw the results off in unpredictable ways.

That's not to say that we can't predict what happens in certain trivial cases (when symmetry can be used to simplify the problem), but there's not a lot of interest in trivial cases (by definition).
It is continuous if you can simplify each attraction down to a 2 body system. Rotation doesn't change anything.
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May 21st, 2017, 03:22 PM   #8
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It is continuous if you can simplify each attraction down to a 2 body system. Rotation doesn't change anything.
What does that have to do with it?
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