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

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Homework Help Overview

The discussion revolves around the Lagrangian mechanics of a free particle, specifically examining the implications of substituting the Lagrangian L with a function f(L) when dL/dt=0. Participants are tasked with showing that this substitution results in the same equations of motion.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss substituting the Lagrangian with f in the Euler-Lagrange equations and applying the chain rule. Questions arise about the validity of concluding that f(L) yields the same equations of motion based on the results obtained.

Discussion Status

Some participants have attempted calculations and shared results, noting that substituting leads to zero in certain cases. There is ongoing exploration of whether this indicates that the same equations of motion are achieved and how to interpret the results regarding the function f'(L).

Contextual Notes

Participants are working under the assumption that the Lagrangian is for a system of free particles and are considering the implications of differentiability and the conditions under which the equations of motion remain unchanged.

nikolafmf
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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?
 
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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. :(
 
Substitute the lagrangian with f in the euler-lagrange equations. Then use chainrule.
 
exclamationmarkX10 said:
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?
 
nikolafmf said:
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).
 
Last edited:

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