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.