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Consider a Randers space $(M,F)$ that is the solution of the zermelo's navigation problem associated to a wind $W$ which is homothety; $\mathcal{L}_Wh=\sigma h$, $\delta$ constant, on a Riemannian space $(M,h)$. Then the Randers geodesics can be found using Theorem 2 of Robles.

Now I am wondering if there is any way with which one can find the Randers geodesics of the Randers space ($\mathbb{R}^3,F$) which comes from putting the wind $W=(b+a\sin kx,0,0)$ on the Euclidean space $\mathbb{R}^3$. By using the equation of the geodesics it sounds quite difficult.

P.S.: here $a, b$ and $k$ are some constant.

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  • $\begingroup$ What main difficulty are you encountering? $\endgroup$ – S.Surace Jun 30 '18 at 6:37
  • $\begingroup$ It demands lot of calculations and one has to deal with a differential equation which is not so easy to be solved. $\endgroup$ – Majid Jul 1 '18 at 0:07
  • $\begingroup$ If it‘s just a lot of work, you could try a CAS. What is the differential equation you are struggling with? $\endgroup$ – S.Surace Jul 5 '18 at 14:21
  • $\begingroup$ @S.Surace What is CAS? I am dealing with an Abel equation of the first kind! $\endgroup$ – Majid Jul 13 '18 at 1:09
  • $\begingroup$ I‘m referring to a computer algebra system, e.g. Mathematica. $\endgroup$ – S.Surace Aug 7 '18 at 7:01

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