How can I solve the for the geodesics of this modified version of the hyperbolic metric inside the closed punctured disc $$ \{ (x,y) \in \mathbb{R} : 0<x^2+y^2\leq 1 \}? $$
$$ ds^2 = \frac{dx^2 + dy^2}{x^2+y^2} $$
I expect the solutions to be two types of geodesics - (Very small) arcs - movement towards the boundary around and back up (ish)...
How can I prove this?