Landau classical field theory question

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Landau derives the relativistic equation of motion for a uniformly accelerated particle by relating the differentials in both frames, specifically using du=dv. He introduces the relationship ds=1/gamma * cdt to connect space and time intervals. The discussion raises questions about the derivation process and the assumptions made in the calculations. Participants express confusion about the steps leading to the differential equation. Clarifying these derivations is essential for understanding the underlying physics.
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One page 24 of his book on classical field theory (4th edition), Landau derives the relativistic equation of motion for a uniformly accelarated particle. How does he get the differential equation that leads him to his result?
 
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I assume it is as follows. He took du=dv in both frames, then said ds=1/gamma * cdt. But that still boters me.
 

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