- #1
danielakkerma
- 231
- 0
Hello everyone!
After having being asked to help prepare a simulation(using computer programming), for the general(numerical) solution to Kepler's orbits(strictly speaking, for any central force, and for simplification, no other is acting here), I ran into a bit of a rut trying to resolve some outstanding issues applying to this problem.
Naturally I am aware of the general solution as it pertains to Polar coordinates, but I've decided to try out solving directly for planar motion on the x,y field.
Here are my equations(and also, please see the diagram attached)
Assuming a central force:
[itex]
\large
\vec{F}=-\vec{\nabla}U = \frac{k}{r^2}
[/itex], Where k is a constant should determine the course of the motion(i.e its being parabolic, hyperbolic, circular, that sort of thing).
In the cartesian coordinate system(the one I use):
[itex]
\large
r^2=x^2+y^2
[/itex]
Which leads to:
[itex]
m\ddot{\vec{r}} = \frac{k}{r^2}\hat{r}
[/itex]
Bifurcating the force to the axes:
[itex]
m\ddot{x} = \frac{k}{r^2}\cos{\phi},
m\ddot{y} = \frac{k}{r^2}\sin{\phi},
\phi = \arctan{\frac{y}{x}}
[/itex]
Naturally these systems have only a numerical solution, and, in doing so, I've received the following output which doesn't make any sense to me, the graph, irrespective of the initial conditions(and of k), leads to conflicting outcomes, which you can evaluate for yourself.
I'll let the images speak for themselves,
I can't thank you enough in advance for your attention to this rather tedious question,
Beholden,
Daniel
P.S
The so-called "ImageA" is plot of (x, y).
After having being asked to help prepare a simulation(using computer programming), for the general(numerical) solution to Kepler's orbits(strictly speaking, for any central force, and for simplification, no other is acting here), I ran into a bit of a rut trying to resolve some outstanding issues applying to this problem.
Naturally I am aware of the general solution as it pertains to Polar coordinates, but I've decided to try out solving directly for planar motion on the x,y field.
Here are my equations(and also, please see the diagram attached)
Assuming a central force:
[itex]
\large
\vec{F}=-\vec{\nabla}U = \frac{k}{r^2}
[/itex], Where k is a constant should determine the course of the motion(i.e its being parabolic, hyperbolic, circular, that sort of thing).
In the cartesian coordinate system(the one I use):
[itex]
\large
r^2=x^2+y^2
[/itex]
Which leads to:
[itex]
m\ddot{\vec{r}} = \frac{k}{r^2}\hat{r}
[/itex]
Bifurcating the force to the axes:
[itex]
m\ddot{x} = \frac{k}{r^2}\cos{\phi},
m\ddot{y} = \frac{k}{r^2}\sin{\phi},
\phi = \arctan{\frac{y}{x}}
[/itex]
Naturally these systems have only a numerical solution, and, in doing so, I've received the following output which doesn't make any sense to me, the graph, irrespective of the initial conditions(and of k), leads to conflicting outcomes, which you can evaluate for yourself.
I'll let the images speak for themselves,
I can't thank you enough in advance for your attention to this rather tedious question,
Beholden,
Daniel
P.S
The so-called "ImageA" is plot of (x, y).