May 15th, 2017, 08:01 AM  #1 
Member Joined: Dec 2016 From: United States Posts: 53 Thanks: 3 Math Focus: Abstract Simulations  The NBody problem, what can we know?
Is there a way to setup an nbody system and know what state it should be via "common sense" or a simple proof that works because the system is nonchaotic? 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, nonchaotic. It's essentially 3 independent 2body problems. Are there other states we can prove at any given T? The above problem I think can be solved with integrals. 
May 15th, 2017, 06:03 PM  #2 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,518 Thanks: 506 Math Focus: Yet to find out. 
No. Chaos will emerge. I one of your other posts, Mashcke highlighted the idea of differential equations applied to a 'predatorprey' system. This is a good starting point. In fact the LotkaVolterra 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. 
May 16th, 2017, 07:58 AM  #3  
Member Joined: Dec 2016 From: United States Posts: 53 Thanks: 3 Math Focus: Abstract Simulations  Quote:
 
May 16th, 2017, 09:03 PM  #4 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,518 Thanks: 506 Math Focus: Yet to find out.  
May 21st, 2017, 05:54 AM  #5 
Member Joined: Dec 2016 From: United States Posts: 53 Thanks: 3 Math Focus: Abstract Simulations 
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 NBody 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 
May 21st, 2017, 10:33 AM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,148 Thanks: 2386 Math Focus: Mainly analysis and algebra 
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 10:35 AM. 
May 21st, 2017, 02:48 PM  #7  
Member Joined: Dec 2016 From: United States Posts: 53 Thanks: 3 Math Focus: Abstract Simulations  Quote:
 
May 21st, 2017, 04:22 PM  #8 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,518 Thanks: 506 Math Focus: Yet to find out.  

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