# Replacing Lagrangian L with function f(L) for free particle

• nikolafmf

## Homework Statement

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If L is Lagrangian for a (system of) free particle(s) and dL/dt=0, show that any twice differentiable function f(L) gives the same equations of motions.

## Homework Equations

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Euler-Lagrange equations.

## The Attempt at a Solution

Well, after some calculation, I get [itеx] $\frac{d}{dt}\frac{\partial f}{\partial \dot{r}}-\frac{\partial f}{\partial r}=0$ [/itеx].

Can I conclude from this that f(L) gives the same equations of motion? If not, what should I do?

Well, in my Latex the command worked as should do. I don't know why in my previous message the equation didn't show up. :(

Substitute the lagrangian with f in the euler-lagrange equations. Then use chainrule.

Substitute the lagrangian with f in the euler-lagrange equations. Then use chainrule.

Thank you for your suggestion. I already did that and got zero as a result. What should I conclude from that?

Thank you for your suggestion. I already did that and got zero as a result. What should I conclude from that?

After that, you should get the same equations of motion except they are multiplied by $f\prime(L)$. You have to then argue that you can divide out the $f\prime(L)$.

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