What is the Hooke's Law Equation for a 3D System with Attached Spheres?

AI Thread Summary
The discussion centers on the application of Hooke's law in a 3D system involving multiple spheres. The main question is whether to express the law as d{2}r{ij}/dt{2} = -[some constant][r{i,j} - r{i,j-1}] or d{2}r{ij}/dt{2} = -[some constant][r{i,j-1} - r{i,j}]. Both formulations create complications when applied to different spheres, leading to inconsistencies in the equations. The complexity of the system risks losing mathematical beauty and symmetry, reminiscent of the classic three-body problem, which, despite its elegance, is simpler than the current scenario. The conversation highlights the challenge of maintaining simplicity and symmetry in complex systems.
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I am working on a theory and this thing is bothering since the past few hours...

When we write down hooke's law that is

d{2}x/dt{2} = -kx

We write down as x as the displacement from the mean position given that the mean position coincides with zero...

Now let's suppose that i have a 3D system.. And a sphere attached with various spheres... Let's say that the sphere we are interested in is (i,j). And out of the several spheres attached to ij let's say we take (i,j-1).

The question is what is my hookes law equation.. Is it

1) d{2}r{ij}/dt{2} = -[some constant][r{i,j} - r{i,j-1}]

or

2) d{2}r{ij}/dt{2} = -[some constant][r{i,j-1} - r{i,j}]

Now it is not that easy.. Cause i need to generalize this.. I have several other spheres attached with (i,j).. And if i use 1 it causes problem with some of the spheres and is okay for the rest... And if i use 2 it causes the same problem...

I can't use 1 for some spheres and 2 for others... Cause then in my 3D infinite space the equations will depend on the position of the spheres...And the equations would be a mess to solve and the most important it loses its beauty...
 
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When things get complicated the mathematical beauty and symmetry has a tendency to fade. In some cases though, new beauty and symmetry can emerge out of complexity. In this case I can't help but think of the classic 3 body problem. I think you can get some simple and describable modes out of such a system, but in general - yea, it gets messy.

Im interested to see some real replies. :p
 
Academic said:
When things get complicated the mathematical beauty and symmetry has a tendency to fade. In some cases though, new beauty and symmetry can emerge out of complexity. In this case I can't help but think of the classic 3 body problem. I think you can get some simple and describable modes out of such a system, but in general - yea, it gets messy.

Im interested to see some real replies. :p

The three body problem is elegant... But in that case there are no complexities involved cause there are just three equations and its easy solving them... So looking at the three body problem wouldn't help...
 
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