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

  1. Dec 27, 2014 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations

    Euler-Lagrange equations.


    3. 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?
     
  2. jcsd
  3. Dec 27, 2014 #2
    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. :(
     
  4. Dec 27, 2014 #3
    Substitute the lagrangian with f in the euler-lagrange equations. Then use chainrule.
     
  5. Dec 28, 2014 #4
    Thank you for your suggestion. I already did that and got zero as a result. What should I conclude from that?
     
  6. Dec 28, 2014 #5
    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].
     
    Last edited: Dec 28, 2014
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