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

  • Thread starter nikolafmf
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  • #1
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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?
 

Answers and Replies

  • #2
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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. :(
 
  • #3
Substitute the lagrangian with f in the euler-lagrange equations. Then use chainrule.
 
  • #4
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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?
 
  • #5
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 [itex]f\prime(L)[/itex]. You have to then argue that you can divide out the [itex]f\prime(L)[/itex].
 
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