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Well, I've done some calculus to your problem. The problem, in fact, is that $|\vec{r_i}-\vec{r_j}|$ everywhere. Also the temporal derivative of that is pain. Of course you can write it without that angle you commented if you write the derivatives in terms of escalar products. However, I'm not going to follow that. You have a system of 3 bodies, isolated, ...

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Other answers and the referenced paper assume a constant radius turn. This path would require a discontinuous steering angle, which is not only non-physical but a somewhat poor approximation for how human drivers drive. The reason drivers don't approximate this technique (besides that it would take very rapid steering wheel movement) is that it would incite ...

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If you plot the path of the moon with respect to the star you should get a cycloid pattern averaging around the path of the planet. That's what our moon does. The planet and the moon are doing a gyrating dance about their center of mass, with the moon doing the most motion (because it has the smallest mass), while the planet/moon duo orbit the star. I've ...

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