When the Lagrangians are equals?

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The discussion centers on a problem from Landau's classical mechanics book regarding the conservation theorem, specifically the relationship between the times taken by particles of different masses but the same potential energy. The solution provided is t'/t = sqrt(m'/m), derived under the assumption that the Lagrangians for both paths are equal (L' = L). The user seeks clarification on the correctness of this assumption and the underlying concept of the Lagrangian in this context.

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I’m not very good with english, it isn’t my native language..., but I’m going to explain my question...

I’m reading the first book of Landau's series ,it’s about clasical mechanics.
In the second chapter you can find a problem about the conservation's theorem

the problem says The first problem says:

Find the ratio of the times in the same path for particles having different masses but the same potential energy.

the solution is: t'/t=sqrt(m'/m)

My tentative solution is supposing that the lagrangian for both paths are the same...

then:

L'=L

1/2m'v'2-U=1/2mv2-U

Finally:

t'/t=sqrt(m'/m)

BUT, It’s that correct?

and why the lagrangians are the same? I’m not sure about the real concept (or meaning) of the lagrangian of a system...

thanks...
 
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It would be helpful if you posted the problem since we don't all have a copy of Landau. Maybe this should be in the homework help section.
 

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