Editing Polar coordinate system (section)

=====Co-rotating frame=====
<!-- This whole section should go to hell -->For a particle in planar motion, one approach to attaching physical significance to these terms is based on the concept of an instantaneous ''co-rotating frame of reference''.<ref name="Taylor">For the following discussion, see {{Cite book |last=John R Taylor |title=Classical Mechanics |publisher=University Science Books |year=2005 |isbn=1-891389-22-X |page=§9.10, pp. 358–359}}</ref> To define a co-rotating frame, first an origin is selected from which the distance ''r''(''t'') to the particle is defined. An axis of rotation is set up that is perpendicular to the plane of motion of the particle, and passing through this origin. Then, at the selected moment ''t'', the rate of rotation of the co-rotating frame Ω is made to match the rate of rotation of the particle about this axis, ''dφ''/''dt''. Next, the terms in the acceleration in the inertial frame are related to those in the co-rotating frame. Let the location of the particle in the inertial frame be (''r''(''t''), ''φ''(''t'')), and in the co-rotating frame be (''r''&prime;(t), ''φ''&prime;(t)). Because the co-rotating frame rotates at the same rate as the particle, ''dφ''′/''dt'' = 0. The fictitious centrifugal force in the co-rotating frame is ''mr''Ω<sup>2</sup>, radially outward. The velocity of the particle in the co-rotating frame also is radially outward, because ''dφ''′/''dt'' = 0. The ''fictitious Coriolis force'' therefore has a value −2''m''(''dr''/''dt'')Ω, pointed in the direction of increasing ''φ'' only. Thus, using these forces in Newton's second law we find:
<math display="block">\mathbf{F} + \mathbf{F}_\text{cf} + \mathbf{F}_\text{Cor} = m\ddot{\mathbf{r}} \, , </math>
where over dots represent derivatives with respect to time, and '''F''' is the net real force (as opposed to the fictitious forces). In terms of components, this vector equation becomes:
<math display="block">\begin{align}
             F_r + mr\Omega^2 &= m\ddot{r} \\
  F_\varphi - 2m\dot{r}\Omega &= mr\ddot{\varphi} \ ,
\end{align}</math>
which can be compared to the equations for the inertial frame:
<math display="block">\begin{align}
        F_r &= m\ddot{r} - mr\dot{\varphi}^2 \\
  F_\varphi &= mr\ddot{\varphi} + 2m\dot{r}\dot{\varphi} \ .
\end{align}</math>

This comparison, plus the recognition that by the definition of the co-rotating frame at time ''t'' it has a rate of rotation Ω = ''dφ''/''dt'', shows that we can interpret the terms in the acceleration (multiplied by the mass of the particle) as found in the inertial frame as the negative of the centrifugal and Coriolis forces that would be seen in the instantaneous, non-inertial co-rotating frame.

For general motion of a particle (as opposed to simple circular motion), the centrifugal and Coriolis forces in a particle's frame of reference commonly are referred to the instantaneous [[osculating circle]] of its motion, not to a fixed center of polar coordinates. For more detail, see [[Centripetal force#Local coordinates|centripetal force]].