I want to plot a solution to the Geodesic equations for the Poincare disk in polar coordinates (though these may be wrong. I want to plot these as a way to check my answer):
$$\ddot{r}+\frac{2r}{1-r^2}(\dot{r})^2+\frac{4r}{1-r^2}\dot{r}\dot{\theta}-\frac{r^2+1}{2}(\dot{\theta})^2 = 0$$
$$\ddot{\theta}-\frac{2}{r(1-r^2)}(\dot{r})^2+\frac{2r^2+1}{r(1-r^2)}\dot{r}\dot{\theta} = 0$$
I solve these equations with NDSolve using some random initial values.
eqns = {r''[t] + (2*r[t])/(1 - r[t]^2)*(r'[t])^2 + (4*r[t])/(
1 - r[t]^2)*(r'[t]*\[Theta]'[t]) - (r[t]^2 + 1)/
2*(\[Theta]'[t])^2 == 0,
\[Theta]''[t] -
2/(r[t]*(1 - r[t]^2))*(r'[t])^2 + (2*r[t]^2 + 1)/(
r[t]*(1 - r[t]^2))*(r'[t]*\[Theta]'[t]) == 0,
r[0] == 0.5, \[Theta][0] == 0,
r'[0] == -0.5, \[Theta]'[0] == 0.5
};
sol = NDSolve[eqns, {r, \[Theta]}, {t, 20}]
Our solution is a curve parametrized by $t$ in polar coordinates. But now I don't know how to plot this curve on the plane (and if someone can help, maybe I can put this curve on top of the unit disk, though that's not my biggest priority). I tried transforming to cartesian then plotting by doing x[t_] = r[t] /. sol + Cos[\[Theta][t] /. sol] but that just returns an error. Any tips on what I should do?
EDIT: These equations are definitely wrong. I think the correct ones are (especially after verifying with some graphs)
$$\ddot{r} + \frac{2r}{1-r^2}(\dot{r})^2-\frac{r(r^2+1)}{1-r^2}(\dot{\theta})^2 = 0$$
and
$$\ddot{\theta}+\frac{2(r^2+1)}{r(1-r^2)}\dot{r}\dot{\theta} = 0$$
