A Finsler perturbation of the Poincaré metric

One method of gaining some insight into Finsler geometry is that of studying small Finsler perturbations of Riemannian metrics. We consider here the standard two-dimensional upper half plane Poincare metric, for which the geodesics are semi-circles and virtical lines. The effect of a simple Finsler perturbation on these geodesics is given by an explicit computation of the perturbed geodesics.