In a 2-body problem is it true that, in all situations, the moving path is closed? In which cases are the paths closed? Fixing the coordinate system or fixing one of the bodies gives us different paths; when are these paths closed?
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Related: Bertrand's theorem on Wikipedia; http://physics.stackexchange.com/q/55638/2451 ; http://physics.stackexchange.com/q/110123/2451 – Qmechanic Jun 16 '14 at 14:57
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Given that the combined center of mass can be moving in a straight line, you have to be careful by what you mean "closed" path. – John Alexiou Aug 25 '14 at 21:43
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The general condition for the orbits to be closed is that the angle $\Phi$ between the pericentre and the apocentre is commensurable with $2\pi$, namely $\Phi/2\pi \in \mathbb{Q}$ i.e. can be expressed as the ratio of two integers. It is easy to see with a computer in the case that this condition is false that the orbit will be dense, namely will go infinitesimally near every point, in the annulus formed with radii the pericentre and the apocentre.
Actually, if you've ever plotted orbits of a two body system, it is almost impossible not to make a mistake that gives you open orbits, because only two classes of potentials always give closed orbits for arbitrary bounded motion for this system, according to Bertrand's theorem. This does not negate the fact that bounded and closed orbits exist for most (all?) potentials, provided that the initial conditions are set appropriately.
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