Invariant Lagrangian

cap.r

1. The problem statement, all variables and given/known data


the question is in the image exactly as i wrote it down in class. but it's basically asking what systems have potential and kinetic energies that form a Lagrangian which is invariant to some transformation X:R^2-->R^2.

2. Relevant equations

The Lagrangian is the only equation I can think of that would be relevant to this. the equation from class is in the image above.

3. The attempt at a solution
in my attempts to find an answer to this I have read a bit of Classical Mechanics by Taylor and have many other books near by that I can refer to. but I am not sure what I am looking for in the index and have yet to find a reasonable answer.
I am also guessing it is somehow related to Neother's Theorem since her theorem tells you that there is a conserved quantity when the Lagrangian is invariant to changes in the coordinates of the system. but as i said I can't put my finger on it.


recommended readings will be appreciated. this is not HW it's a challenge question by the prof. and I am just looking for an answer since the question is intriguing.

AEM

You are right that your problem is related to Noether's theorem. But of more use to you than my suspicions, will be section 3.2 of Classical Dynamics, A Contemporary Approach by Jorge Jose and Eugene Saletan. Hopefully you can locate a copy because page 120 deals explicity with how a Lagrangian transforms. By the way, if you're a physics major and if you can afford that book, it's very good.

cap.r

on the way to the library to find that book... anyone else have suggestions or leads that l can fallow?
thanks