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Newton first determined that gravity weakens as the inverse square of the distance. What would happen if gravity obeyed a different force law? It is an amazing fact that almost any other force law leads to orbits that do not "close" or come back to the places that they started with the original velocities. In fact only gravity (force proportional to inverse distance squared) and the harmonic oscillator (force proportional to distance) lead to closed orbits for all initial conditions. Furthermore, although all central forces allow circular orbits, these orbit are not always stable (see figure). You can explore different orbits and different force laws yourself with the Central Force Integrator.
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