I am a little confused about the condition for circular orbit. Goldstein's Classical Mechanics has the condition for circular orbit as $$f'=0\tag1$$ where $f'$ is the effective force. I understand that the reason for this requirement is that the corresponding $V'$is at an extremum. However, setting $f'=0$ yields $$f(r)=-\frac{l^2}{mr^3}\tag2$$ and the one of the equations of motion becomes $$m\ddot r=0\tag3$$ This can't be right as it is saying that the net force on the object is zero, and yet the object is supposedly in a circular orbit. Am I interpreting equation (3) wrong?
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In the coordinate frame where the satellite is at rest, there is also a centrifugal force which balances the central force.
answered Aug 8, 2012 at 9:23
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$\begingroup$ Ok, but I have a follow up question. The equation of motion in particular was derived after showing that the 2 body central force motion problem can be reduced to an equivalent one body problem, with one body fixed at the center of force. One would then expect the equation of motion(EOM) to describe the motion of the non-fixed body i.e., the EOM should be from the perspective of the fixed body, which in this case is most naturally the Earth. The EOM in question is: $$m \ddot r - \frac{l^2}{m{r^3}}=f(r)$$ $\endgroup$– JoebevoAug 10, 2012 at 3:25
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$\begingroup$ I see no question mark, but here is what I think you are asking. Let us for simplicity assume that both objects $M$ and $m$ are point-like, and that the mass ratio $M/m$ between the two objects is infinite, so that we can ignore movement of the larger object $M$. To fix the frame of reference, pick the origin at $M$'s position, and pick two axes in the orbit plane so that one of them goes through $m$. This coordinate frame is non-inertial, so there will appear fictitious forces. $\endgroup$– Qmechanic ♦Aug 10, 2012 at 14:46
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$\begingroup$ Sorry, my question, to put it as simply as possible, was: Why is it that the equation of motion is being written in the coordinate frame in which the satellite is at rest? In a situation like this, one usually expects the equation to be written in the coordinate frame where the $\it earth$ is at rest. $\endgroup$– JoebevoAug 12, 2012 at 12:03
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$\begingroup$ Consider the non-inertial coordinate system defined in my previous remark. $M$ is at rest by definition. If $m$ is performing a circular orbit (relative to an inertial frame), it will also be at rest. Hence both objects are at rest (as seen from the non-inertial coordinate frame). $\endgroup$– Qmechanic ♦Aug 12, 2012 at 12:21
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$\begingroup$ Ok, it looks like I have to learn about non-inertial coordinate frames to understand this. Thanks anyway. $\endgroup$– JoebevoAug 14, 2012 at 5:10