Lagrangians and Noether Theorem

1. Jan 13, 2013

atomqwerty

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

Let be the lagrangian given by

$L(x,y,\dot{x},\dot{y})=\frac{m}{2}(\dot{x^2} +\dot{y^2})-V(x^{2}+y^{2})$

and

$L(x,y,\dot{x},\dot{y})=\frac{m}{2}(\dot{x^2} + \dot{y^2})-V(x^{2}+y^{2}) - \frac{k}{2}x^{2}$

and the transformation

$x'=\cos\alpha x - \sin\alpha y$
$y'=\sin\alpha x + \cos\alpha y$

Show the invariant observable with respect to the symmetry of this transformation (Noether's Theorem) for both Lagrangians.

2. Relevant equations

Noether Theorem says that for each variable x such as $\partial L/\partial x = 0$ then x is said to be invariant and $\partial L/\partial \dot{x} = 0$ is constant of motion for that system.

3. The attempt at a solution

First, the transformation from (x,y) to (x',y') doesn't change the Lagrangian, and since L does depend on $x, y, \dot{x}, \dot{y}$ on both Lagrangians, then the only non explicit variable is time t, and in that case the conserved quantity would be the Energy E = T + V. I must be wrong since this is a problem of an exam of Theoretical Mechanics in University...

Thanks a lot!

Last edited by a moderator: Jan 19, 2013
2. Jan 13, 2013

dextercioby

You need to think about Noether's theorem. Can you show that the 2 Lagrangians are invariant under the mentioned transformations ?

3. Jan 13, 2013

atomqwerty

I could check that both $\dot{x^2}+\dot{y^2}$ and $x^2+y^2$ keep the same under that transformation, and so that there's no variable $x$ or $y$ that verifies $\partial L / \partial x =0$, $\partial L / \partial y =0$. That's what I thought of time instead...

4. Jan 13, 2013

physicus

Noether's theorem is something else. It states, that for every continuous symmetry of a theory there is a conserved charge. You simply try to find cyclic coordinates which does not give you a conserved quantity here.

What you need to do is to consider an infinitesimal transformation. So just choose $\alpha$ infinitesimal. Therefore:
$x'=x-\alpha y, y'=y+\alpha x$

The conserved quantity is then given by
$\frac{\partial L}{\partial \dot{x}}\frac{\delta x}{\alpha}+\frac{\partial L}{\partial \dot{y}}\frac{\delta y}{\alpha}$

You will see, for example, that it is angular momentum $m(\dot{y}x-\dot{x}y)$ for the first lagrangian.

5. Jan 13, 2013

atomqwerty

And by $\delta x$ do you mean $-\alpha y$, and by $\delta y$ do you mean $\alpha x$?

6. Jan 13, 2013

physicus

Yes, exactly.

7. Jan 19, 2013

atomqwerty

Why is the conserved quantity given by that expression? I've just been thinking about it via Poisson brackets and I can't derive it. Thanks

Last edited: Jan 19, 2013
8. Jan 19, 2013

physicus

The Lagrangian is invariant under the transformation $\delta L=0$.

I just prove the conservation of the Noether charge for one variable x instead of x and y. The proof obviously works for more variables:
You want to show $\frac{\partial L}{\partial \dot{x}}\frac{\delta x}{\alpha}$ is conserved, i.e. $\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{x}}\frac{\delta x}{\alpha}\right)=0$

$\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{x}}\frac{\delta x}{\alpha}\right) = \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right)\frac{\delta x}{\alpha}+\frac{\partial L}{\partial \dot{x}}\frac{\delta \dot{x}}{\alpha}=\frac{\partial L}{\partial x}\frac{\delta x}{\alpha}+\frac{\partial L}{\partial \dot{x}}\frac{\delta \dot{x}}{\alpha}= \frac{1}{\alpha}\delta L = 0$

In the second step I have used the Euler Lagrange equations to simplify the first term. This is actually important. The conserved charge is only conserved for solutions of the Euler Lagrange equations. Classically, all particles obey the equations. However, in QFT for example, there are so-called off-shell particles, that do not obey the equations of motion. For these, the Noether current is not necessarily conserved.

9. Jan 20, 2013

atomqwerty

I can see that the same quantity )angular momentum) is conserved also for the second Lagrangian, isn't it?

10. Jan 20, 2013

physicus

The second Lagrangian is not invariant under the given transformation. Are you sure it is right?

11. Jan 21, 2013

atomqwerty

Well, following
then the new term of the second Lagrangian does not depend on $\dot{x}$ or $\dot{y}$, right?

12. Jan 21, 2013

physicus

This formula for the conserved charge is only valid if the Lagrangian is invariant under the given transformation (you have seen that I used $\delta L=0$ in the proof). But I don't see why the second Lagrangian should be invariant. Pick for example $\alpha=\frac{\pi}{2}$, then $x \rightarrow -y, y \rightarrow x$. Then the tranformed Lagrangian is
$\frac{m}{2}(\dot{x^2} + \dot{y^2})-V(x^{2}+y^{2}) - \frac{k}{2}y^{2} \not= L$. There seems to be an error in the problem the way you have written it down in the beginning.