Elliptic Function Rotation Problem

  • #1
29
2

Summary:

I have made a rotating point on the perimeter of an ellipse. My problem is that the motion is reversed from the true foci.
Hi all:)
In my recent exploration of Elliptic Function, Curves and Motion I have come upon a handy equation for creating orbital motion.
Essentially I construct a trigonometric function and use the max distance to foci as the boundary for my motion on the x-plane.
Desmos Orbit physicsforum2.png


When I plot a point rotating around the perimeter of my Ellipse I get my desired changing velocity depending on the distance to the foci; shown in this:
https://gyazo.com/9430d22ff4d6f38f2d5bcf381a06db76

However it appears that the point is rotating faster near the ambiguous foci and not the true foci. How can I reverse my function so that the point will move faster near my (0,0) coordinate and slower when it moves further away, instead of the opposite?
 

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  • #2
270
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Summary:: I have made a rotating point on the perimeter of an ellipse. My problem is that the motion is reversed from the true foci.

Hi all:)
In my recent exploration of Elliptic Function, Curves and Motion I have come upon a handy equation for creating orbital motion.
Essentially I construct a trigonometric function and use the max distance to foci as the boundary for my motion on the x-plane.
View attachment 269515

When I plot a point rotating around the perimeter of my Ellipse I get my desired changing velocity depending on the distance to the foci; shown in this:
https://gyazo.com/9430d22ff4d6f38f2d5bcf381a06db76

However it appears that the point is rotating faster near the ambiguous foci and not the true foci. How can I reverse my function so that the point will move faster near my (0,0) coordinate and slower when it moves further away, instead of the opposite?
I am not quite clear on what you are trying to simulate with your program. However, if you want to simulate Kepler elliptical orbital motion, for example a comet orbiting the sun, then the book by Landau "Classical Mechanics" gives the solution to the problem in polar coordinates. I assume you have knowledge of calculus. Landau's solution uses polar coordinates show in the image:
elipse.jpg

where ##a## is the semi-major axis, ##b## is the semi-minor axis, ##r## is the distance from the center of the ellipse to a point on its perimeter, and ##\xi## is the angle between ##r## and the semi-minor axis.

Landau derives the equations for Kepler orbital motion to be:$$
r=a(1-e\cos(\xi))
$$
$$
t=\sqrt(\frac{ma^3}{\alpha})(\xi -e\sin(\xi) )
$$
where ##e## is the eccentricity of the ellipse, ##t## is time, ##m## is the reduced mass of the system of two masses and ##\alpha## is the strength of the potential field. Your problem, as I see it, is to find a function the describes the speed ##|v|## of the orbiting mass as a function of the angle ##\xi##. To this end I claim,
$$
|v|= \sqrt(\dot{\xi}^2 + \dot{r}^2 )
$$
where the dot above the variable indicates differentiation w.r.t time. We find,
$$
\dot{r}=e\dot{\xi}\sin(\xi)
$$
To find ##\dot{\xi}## we differentiate ##t## w.r.t ##\xi##,
$$
\frac{dt}{d\xi}= \sqrt(\frac{ma^3}{\alpha})(1-e\cos(\xi))
$$
and therefore,
$$
\dot{\xi}=\frac{1}{\sqrt(\frac{ma^3}{\alpha})(1-e\cos(\xi))}
$$
After some algebra we find the speed as a function of ##\xi##,
$$
|v|=\frac{\sqrt(1+e^2 \sin^2 (\xi))}{\sqrt(\frac{ma^3}{\alpha})(1-e\cos(\xi))}
$$
 

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