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It is an exercise from an textbook of Classical Mechanics. Specifically a chapter about calculus of variations. I would like a hint first. What I don't understand is how to apply Euler equation when $y(x)$ is defined by parts (is it a function $y_1$ in the first media and $y_2$ in the second?).

John Douma
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Carlos Oliveira
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    Draw a picture. Although the speed of light is different in each medium you can derive the time spent in each medium. From that you minimize the total time and you are done. If you search the internet you should be able to find this derivation. – John Douma Jun 23 '19 at 05:42
  • I have obtained two integrals. $\int_{y_1}^{y_2} \dfrac{dy}{sin \theta_1 v_1} + \int_{y_2}^{y_3} \dfrac{dy}{sin \theta_2 v_2}$. How do I minimize two integrals with Euler's equation? I have been thinking about merging them, but I didn't succeed. – Carlos Oliveira Jun 23 '19 at 15:31

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