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Suppose that [tex]R^2[/tex] is provided with the following metric

[tex]

ds^2 = dx^2 + (\cosh(x))^2 dy^2

[/tex]

Can we find a general exact formula [tex]\alpha(t)[/tex] for the geodesics (starting at an arbitrary point) ?

The geodesic equation gives

[tex]

x'' - \cosh(x)\sinh(x) (y')^2 = 0

[/tex]

[tex]

y'' + 2 \tanh(x) x' y' = 0

[/tex]

I guess that since this model is simply a reparametrization of the Hyperbolic space on R^2 the geodesics should be known ?

Thank you