When Are Lagrangians the Same?

[rmadsanmartin]

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

Homework Statement

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)

Homework Equations

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The Attempt at a Solution

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...

 

[rmadsanmartin]

This is the section of Landau (about particles and potential energy assuming U is a homogeneous function)

 

[TSny]

Hello.

In the statement of the problem, the potential energy is not assumed to be a homogeneous function of the coordinates. So, I don't think that the material on the page that you attached is directly relevant to this problem.

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

The Lagrangians for m and m' are not the same when using the same time parameter in both Lagrangians (because of the difference in mass). Using the assumption that the potential energy is the same for both masses, see if you can transform the Lagrangian for m' into the Lagrangian for m by re-scaling the time for the m' system. [EDIT: This might be what you did essentially. I'm not sure.]