Geodesics on R^2

  1. Hello,

    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
     
  2. jcsd
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