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?

 

[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).