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0

Rotation is absolute. And in any non-rotating reference frame, the earth and the sun (ignoring all the other bodies in the solar system, and the rest of the universe) both revolve around their common center of mass. Since the mass of the sun is so much greater than the mass of the earth, it is pretty close to saying that the sun is stationary and the earth ...

3

For your second case, you can change the angular momentum, but remember that you have fixed your total energy. You can't make the planet revolve arbitrarily fast or it will have more energy than allowed. By increasing the angular momentum without adding energy, you are circularizing the orbit. To add, you might take a look at the Specific Orbital Energy ...

5

I assume you're talking about the numerical instabilities that arise from having an infinite potential at $r=0$. Here are three common solutions: Use a soft-core potential that behaves like $1/r$ except very close to $r=0$ where it levels off to a finite value. For example, $1/\sqrt{\epsilon+r^2}$ instead of $1/r$ is common. Add hard sphere collision ...

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