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Left: Shows two balls of equal mass suspended from a ceiling with string, Right: Shows to large orbiting bodies in outer space

When I was taking classical mechanics, we dealt a lot with pendulums, and orbiting bodies problems. This lead me to think about the two situations depicted above. Left: Shows two balls of equal mass suspended from a ceiling with string, Right: Shows to large orbiting bodies in outer space.

For some reason, my intuition is begging that there these two cases are somehow mathematically related. However, I need help finding this relationship. It seems that the as the string twists and the balls get closer and closer to each other, its awfully similar to the case when two bodies in space orbit each other until they collide too! (Or am I just crazy and thinking too much?)

What relevant equations / geometric relationships are needed here to establish a relationship between these two cases? (Kepler's Laws, what else?)

enter image description here Going further: Given that the strings magically "go through each other" at every rotation, can't this be related to the case in which the two orbiting masses orbit a third body, say a star, located at the center of the axis of rotation? (Picture not provided here, but I hope you understand my further question)

Again, What relevant equations are needed here to establish a relationship between these two cases? enter image description here Going even further: This is getting pretty hand wavy, but I was thinking that perhaps this could be a new way of looking at the 3-body problem or multi-body problem using the string and ball model? Would Knot theory come into play?

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One potential problem, in your first diagram, they'll describe circular (well spiral if no magic strings!) orbits, there seems to be no scope for elliptical orbits like you'd want for inverse square law. –  twistor59 Jan 3 '13 at 17:31
    
@twistor59 they'll be ellipses if their initial motion is non circular... just like orbits. Angular momentum and energy are independent parameters. –  zhermes Feb 2 '13 at 20:42
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The forces are different, so I don't think the equations will turn out to be the same.

In the second problem you have just gravitational force between the two bodies : $$ F = G\frac{m_1m_2}{r^2} $$ in the pendulum problem you have the "centrifugal force" and the force of the gravity of the earth acting in perpendicular directions balanced by string tension. The rate of the spheres coming together will likely be different, I don't have time to prove it at the moment however...

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Two bodies orbit each other in space will not collide.

So, these two problems are not similar.

The only thing they have in common is angular momentum conservation law.

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It's not entirely clear what aspects you're trying to relate to eachother... are you trying to describe a steady state phenomenon in the penduli example (i.e. some sort of magical strings that keep lengthening?). Both systems are examples of (simple) harmonic motion. That's basically as far as the similarity goes. The twisting penduli aren't effected by the parameters of the other mass - the deciding characteristics are the strings themselves. It's also a little awkward to make an analogy to gravity, with a system that itself is dependent on gravity. Maybe you'd have more luck with a spring system?

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