When we have two bodies and a central force acting towards the center of each other, we could treat the whole problem as a one body problem by introducing the relative coordinate. My question is, when you were to calculate the relative coordinate, you are working out the acceleration of a fictitious particle, which does not exist.
However, if you work out the coupled differential equations, you would get exact acceleration of both particles.
Therefore I think reduced mass is not a reliable method , but I was told that the reduced mass method can get the exact solution as the coupled differential equation. Therefore I think I am wrong, could anyone shed the light on this please?
Answer
Reducing a two body problem into a effective one body problem is just an mathematical manipulation to simplify the coupled differential equations, that you have to solve, and convert them to an easier problem (two uncoupled differential equations). There is no approximation involved. Actually, it is just a transformation of variables and after you have solved the problem in these variables you can transform back and you have the exact solution of your original problem. The concept of a reduced mass is only introduced, because after the transformation the differential equation of the relative coordinate looks like one of a fictitious particle with this particular mass.
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