gravity on fractals

[Tglad]

I started with a simple case, does anyone know how this cluster of planets will move if starting from stationary?

[Tglad]

well the pull to the right of the 1kg planet is:
 2/2² + 4/6² + 8/14² + ...
and the pull to the left is:
 0.5/1² + 0.25/1.5² + 0.125/1.75² + ...
So both sides have a pull of:
 0.5 + 0.1111 + 0.0408 + ...
So the 1kg planet has total force of 0 and doesn't move.
The same is true of every other planet, so no planets move... which is surprising.

Another way to think of it is that the planets are an infinite evenly spaced set of equal mass planets (so this wouldn't move) that have had a transform applied, so planets on the left are brought closer but reduced in mass to compensate, and planets to the right are spaced further but increased in mass to compensate. So it is a transformed tiling.

I think you can transform both ends too, to give this solar system which stays put without orbitting... fractal planets are unusual!

[cbuchner1]

it's really hard to see why this would not start collapsing towards the center. After all there is no initial kinetic energy stored in the system to prevent that.

But if it's an equilibrium, as you demonstrated by calculation - it would be an extremely unstable one, right? One butterfly flaps its wing and the thing comes crashing down.

[Tglad]

It certainly looks like it would collapse inwards, but every planet has equal force pulling left and right, as in picture below.
I think you're right that it is an unstable balance... but not certain.

[cKleinhuis]

ehrm, sorry, but the last 2 pictures would collapse !
directly to the center!

[cKleinhuis]

and i see no evidence for that the planets on the first image wouldnt move

why does the left 1kg planet has a pull to the left ?!?!?!?

[Tglad]

Because to the left is a 0.5kg planet 1m away (so has a force of 0.5/1²) and left of that is a 0.25kg planet 1.5m away.
I didn't write the number of kg on all planets, I hope you can see that it is an infinite series that doubles in mass and doubles in distance.

The strength of gravity decreases with the square of its distance away.

As I drew on the last picture, the planet has the pull of large masses to the left (but quite far away) and small masses to the right (but very close), so the idea is that mass/dist² is equal on both sides of every planet... since none of them have a resulting force, none will have an acceleration, so won't move.

On second thoughts, it does seem like its a trick. A bit like a levitating turtle because he's sitting on a smaller turtle which is in turn sitting on a smaller turtle etc.
So I wonder what the motion really is, or if it can really be calculated given the infinite number of bodies.

[cKleinhuis]

you are assuming endless amounts of planets in both directions ?
i still wonder if they wouldnt move

at least it could be simulated with such an amount of planets would yield a rounded 0 gravity at some distance, and it would be really interesting to see what happens when
an butterfly comes along

but beside of that, i thought about such a set up could be the start of our universe ?

and i know that gravity produces large scale accelerations on the movements, so, in the end the whole system would explode...

i once wrote a gravity particle simulator... and i wanted to create randomly orbits, or equillibrium systems like our solar system,
but i gave up because it was really hard to do it with random initial setups, although, some small orbits where visible, but
they soon collapsed.

[jehovajah]

Interesting as usual Tom.
This is a case of unstable dynamic equilibrium. I think Steven Hawkings did the calculations with his students for his programme Steven Hawkings universe.

Do you know of the "paradox" of the athlete and the tortoise? The tortoise moves exactly one tenth of the distance the athlete does. Can he ever pass the tortoise if he starts 10 stadia behind him?

Well, yes in real life we regularly beat tortoises in races! The lesson here is to free your mind from whatever constraints imposed on it and think outside the box!

At the "time" this question was posed Europeans had no concept of time as we do today. In fact it was quite common to denote what we call time by a distance. This is principally what Gallileo was attempting to establish in science: a  scientific convention of time. The period of a pendulum was eventually adopted for this purpose and Huygens really did become the father of scientific time!

Can we please give the poor butterfly a break? He didn't do it! It was a cilia on the outside of a bacteria!!