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corindo
corindo
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The equations of the phase curves in the phase portrait of the simple plane pendulum actually correspond to different energy conservation relations: $$ \dot{\theta}^2 - \frac{g}{l}\cos(\theta) = C_0 $$

And in the colored graph of $\sin(z)$ in the complex plane the lines are the lines of constant magnitude: $$ \sin(x+yi) = C $$$$ \|\sin(x+yi)\|^2 = C $$ which can be transformed into another form by the steps below $$ \begin{align} \|\sin(x)\cosh(y) + i\cos(x)\sinh(y)\|^2 &= C \\ \sin(x)^2\cosh(y)^2 + \cos(x)^2\sinh(y)^2 &= C \\ (\sin(x)^2 + \cos(x)^2)\frac{e^{2y}+e^{-2y}}{2} + \sin(x)^2-\cos(x)^2 &= C \\ \frac{e^{2y}+e^{-2y}}{2} -\cos(2x) &= C \end{align} $$ when $y$ is not far from $0$, $\frac{e^{2y}+e^{-2y}}{2} \approx 4y^2 = (2y)^2$,so if we replace $(x,y)$ by $(u,v)$ with $u=2x, \, v=2y$, then the equation becomes $$ v^2 -\cos(u) = C. $$ I think this is why the two plots look so similar. When $y$ goes far from $0$, their forms may no longer be such similar.

The equations of the phase curves in the phase portrait of the simple plane pendulum actually correspond to different energy conservation relations: $$ \dot{\theta}^2 - \frac{g}{l}\cos(\theta) = C_0 $$

And in the colored graph of $\sin(z)$ in the complex plane the lines are the lines of constant magnitude: $$ \sin(x+yi) = C $$ which can be transformed into another form by the steps below $$ \begin{align} \|\sin(x)\cosh(y) + i\cos(x)\sinh(y)\|^2 &= C \\ \sin(x)^2\cosh(y)^2 + \cos(x)^2\sinh(y)^2 &= C \\ (\sin(x)^2 + \cos(x)^2)\frac{e^{2y}+e^{-2y}}{2} + \sin(x)^2-\cos(x)^2 &= C \\ \frac{e^{2y}+e^{-2y}}{2} -\cos(2x) &= C \end{align} $$ when $y$ is not far from $0$, $\frac{e^{2y}+e^{-2y}}{2} \approx 4y^2 = (2y)^2$,so if we replace $(x,y)$ by $(u,v)$ with $u=2x, \, v=2y$, then the equation becomes $$ v^2 -\cos(u) = C. $$ I think this is why the two plots look so similar. When $y$ goes far from $0$, their forms may no longer be such similar.

The equations of the phase curves in the phase portrait of the simple plane pendulum actually correspond to different energy conservation relations: $$ \dot{\theta}^2 - \frac{g}{l}\cos(\theta) = C_0 $$

And in the colored graph of $\sin(z)$ in the complex plane the lines are the lines of constant magnitude: $$ \|\sin(x+yi)\|^2 = C $$ which can be transformed into another form by the steps below $$ \begin{align} \|\sin(x)\cosh(y) + i\cos(x)\sinh(y)\|^2 &= C \\ \sin(x)^2\cosh(y)^2 + \cos(x)^2\sinh(y)^2 &= C \\ (\sin(x)^2 + \cos(x)^2)\frac{e^{2y}+e^{-2y}}{2} + \sin(x)^2-\cos(x)^2 &= C \\ \frac{e^{2y}+e^{-2y}}{2} -\cos(2x) &= C \end{align} $$ when $y$ is not far from $0$, $\frac{e^{2y}+e^{-2y}}{2} \approx 4y^2 = (2y)^2$,so if we replace $(x,y)$ by $(u,v)$ with $u=2x, \, v=2y$, then the equation becomes $$ v^2 -\cos(u) = C. $$ I think this is why the two plots look so similar. When $y$ goes far from $0$, their forms may no longer be such similar.

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corindo
corindo
  • 3.8k
  • 16
  • 30

The equations of the phase curves in the phase portrait of the simple plane pendulum actually correspond to different energy conservation relations: $$ \dot{\theta}^2 - \frac{g}{l}\cos(\theta) = C_0 $$

And in the colored graph of $\sin(z)$ in the complex plane the lines are the lines of constant magnitude: $$ \sin(x+yi) = C $$ which can be transformed into another form by the steps below $$ \begin{align} \|\sin(x)\cosh(y) + i\cos(x)\sinh(y)\|^2 &= C \\ \sin(x)^2\cosh(y)^2 + \cos(x)^2\sinh(y)^2 &= C \\ (\sin(x)^2 + \cos(x)^2)\frac{e^{2y}+e^{-2y}}{2} + \sin(x)^2-\cos(x)^2 &= C \\ \frac{e^{2y}+e^{-2y}}{2} -\cos(2x) &= C \end{align} $$ when $y$ is not far from $0$, $\frac{e^{2y}+e^{-2y}}{2} \approx 4y^2 = (2y)^2$,so if we replace $(x,y)$ by $(u,v)$ with $u=2x, \, v=2y$, then the equation becomes $$ v^2 -\cos(u) = C. $$ I think this is why the two plots look so similar. When $y$ goes far from $0$, their forms may no longer be such similar.