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

[nikolafmf]

Homework Statement

[/B]
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

[/B]
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?

 

[nikolafmf]

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. :(

 

[exclamationmarkX10]

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

 

[nikolafmf]

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?

 

[exclamationmarkX10]

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 $f\prime(L)$.