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I want to plot the following

$$L(\rho) = m + \frac{c}{\rho^2}$$

That's very straightforward.

y6[m_, c_, \[Rho]_] := m + c/\[Rho]^2 
Plot[{y6[1, -2, r], y6[1, -0.5, r], y6[1, -0.1, r], y6[1, 0, r], 
  y6[1, 0.1, r], y6[1, 0.5, r], y6[1, 2, r]}, {r, 1, 10}, 
 PlotLegends -> "Expressions", PlotRange -> {0.76, 1.25}, 
 AxesOrigin -> {1.2, 0.84}]

The ($L$ vs $\rho$)-plot is

The plot

A valid change of coordinates is the following

$$\rho = r \sin(\theta)$$ $$L = r \cos(\theta)$$

And the corresponding plot is the $r \cos(\theta)$ vs $r \sin(\theta)$.

I don't know how to implement correctly the change of coordinates and how to write the command to get the $r \cos(\theta)$ vs $r \sin(\theta)$ plot.

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Perhaps you are looking for this?

ParametricPlot[Evaluate@Table[{r Cos[θ], r Sin[θ]}, {r, 1, 10}],
 {θ, 0, 2 Pi}, PlotLegends -> Array["r = " <> ToString[#] &, 10]]

enter image description here

Edit

Some overkill here, but out of time.

m = 1;
ParametricPlot[Evaluate@Flatten[
   Table[{r Sin[θ], m + c/(r Sin[θ])^2}, {r, 1, 10},
    {c, {-2, -0.5, -0.1, 0, 0.1, 0.5, 2}}], 1], {θ, 0, 2 Pi},
 AxesLabel -> {"r Sin[θ]", "r Cos[θ] = m + c/(r Sin[θ])^2"},
 ImageSize -> 550, BaseStyle -> 12, 
 PlotStyle -> Array[ColorData[97], 7],
 PlotLegends -> Placed[Map["c = " <> ToString[#] &,
    {-2, -0.5, -0.1, 0, 0.1, 0.5, 2}], Right],
 AspectRatio -> 1/GoldenRatio]

enter image description here

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  • $\begingroup$ That is helpful indeed. Thank you. I am wondering if and how it is possible to include the specific relation between $r \sin u$ and $r \cos u$. That is using, $r \cos u = m + c/(r \sin u)^2 $ and plot this particular relation with $r \cos u$ in the y-axis and $r \sin u$ in the x-axis. $\endgroup$ – Konstantinos Feb 25 '18 at 14:12
  • $\begingroup$ Thank you for the added bit. Great reply. Cheers!!! $\endgroup$ – Konstantinos Feb 25 '18 at 14:58

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