# Geodesics on R^2

by Statis
Tags: geodesics
 P: 1 Hello, Suppose that $$R^2$$ is provided with the following metric $$ds^2 = dx^2 + (\cosh(x))^2 dy^2$$ Can we find a general exact formula $$\alpha(t)$$ for the geodesics (starting at an arbitrary point) ? The geodesic equation gives $$x'' - \cosh(x)\sinh(x) (y')^2 = 0$$ $$y'' + 2 \tanh(x) x' y' = 0$$ I guess that since this model is simply a reparametrization of the Hyperbolic space on R^2 the geodesics should be known ? Thank you

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