Does this work?

[uncool]

I'm trying to find some general things about orbits in gravitational systems in multiple dimensions. I've already found that it has to all be in one plane (easy enough), and now I'm assuming the following:

[math]F = \frac{g*m_1*m_2}{d^{#dim-1}}[/math]

[math]F_{x} = F*\frac{x}{d}[/math]

[math]F_{y} = F*\frac{y}{d}[/math]

So:

[math]\frac{d^{2}y}{y*dt^2}=\frac{d^{2}x}{x*dt^2}[/math]

[math]Let |x| = e^{f(t)}[/math]

[math]|y|=e^{g(t)}[/math]

[math]\frac{dx}{dt}=f'(t)e^{f(t)}[/math]

[math]\frac{d^{2}x}{dt^{2}}=f''(t)e^{f(t)}+f'(t)^{2}e^{f(t)}[/math]

[math]\frac{dy}{dt}=g'(t)e^{g(t)}[/math]

[math]\frac{d^{2}y}{dt^{2}}=g''(t)e^{g(t)}+g'(t)^{2}e^{g(t)}[/math]

So:

[math]f''(t)+f'(t)^{2}=g''(t)+g'(t)^{2}[/math]

[math]f''(t)-g''(t)=g'(t)^{2}-f'(t)^{2}[/math]

[math]\frac{(f'(t)-g'(t))'}{f'(t)-g'(t)}=-f'(t)-g'(t)[/math]

[math]ln|f'(t)-g'(t)|=-f(t)-g(t)+C[/math]

[math]f(t)=ln(|x|),g(t)=ln(|x|)[/math]

[math]f'(t)=\frac{x'}{x},g'(t)=\frac{y'}{y}[/math]

[math]|\frac{x'}{x}-\frac{y'}{y}|=C/|xy|[/math]

[math]|x'y-y'x|=C[/math]

Since it is constant, let us assume that x'y is always greater.

[math]x'y-y'x=C[/math]

Can anything happen from there?

-Uncool-