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Question: how do we get the euler lagragian that is written...

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In vacuum the speed of light is a constant so v(x,y) The Euler Lagrange equation is Integrating this gives constant The only solution of this is constant, ni or integrating
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How do we get the Euler lagragian that is written there. I know the formula for it but I don’t get how you arrive at what’s written. I’ve included what the lagrangian is given by the book.
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