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 finding the time ratio for particles of different masses but equal potential energy. The proposed solution is t'/t=sqrt(m'/m), derived from the assumption that the Lagrangians for both paths are the same. The user seeks clarification on whether this assumption is correct and the meaning of the Lagrangian in this context. Additionally, there is a suggestion to post the problem for better understanding, and a note that this inquiry might fit better in a homework help section. Understanding the Lagrangian's role is crucial for solving such mechanics problems effectively.
rmadsanmartin
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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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