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Invariant Lagrangian

by cap.r
Tags: invariant, lagrangian
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cap.r
#1
Jan27-09, 02:05 PM
P: 68
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.
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AEM
#2
Jan27-09, 09:42 PM
PF Gold
P: 362
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
#3
Jan29-09, 11:11 AM
P: 68
on the way to the library to find that book... anyone else have suggestions or leads that l can fallow?
thanks


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