Systems with Lagrangians invariant under coordinate transformations

[cap.r]

Homework Statement

http://img261.imageshack.us/img261/5923/14254560bc0.th.jpg

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.

Homework 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.

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