# Invariant Lagrangian Homework: Find Solutions

• cap.r
In summary, the conversation discusses a question about systems with potential and kinetic energies that form a Lagrangian which is invariant to a transformation X:R^2-->R^2. The equation from class is shown in the image provided. The solution to this problem is related to Noether's theorem, which states that there is a conserved quantity when the Lagrangian is invariant to changes in the coordinates of the system. A recommended reading, section 3.2 of Classical Dynamics by Jorge Jose and Eugene Saletan, is suggested for further understanding of how a Lagrangian transforms. The person asking for help is heading to the library to find the book, and any other suggestions or leads are appreciated.
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.

Last edited by a moderator:
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.

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

## 1. What is an invariant Lagrangian?

An invariant Lagrangian is a mathematical tool used in classical mechanics to describe the motion of a system. It is a function that remains unchanged regardless of the reference frame or coordinate system used to describe the motion.

## 2. How is an invariant Lagrangian used?

An invariant Lagrangian is used to determine the equations of motion for a system. By finding the solutions to these equations, we can predict the future behavior of the system.

## 3. What are the advantages of using an invariant Lagrangian?

One advantage is that it simplifies the equations of motion, making them easier to solve. Additionally, it allows for the use of different coordinate systems, making it more versatile than other methods.

## 4. Are there any limitations to using an invariant Lagrangian?

One limitation is that it can only be used for systems that are conservative, meaning that no energy is lost or gained during the motion. It also does not take into account any external forces acting on the system.

## 5. How do I find solutions for an invariant Lagrangian homework problem?

To find solutions, you will need to use calculus and apply the principles of classical mechanics. You will need to set up and solve the equations of motion derived from the invariant Lagrangian. It may also be helpful to refer to your textbook or seek assistance from a tutor or classmate.

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