# Construct the Lagrangian for the system

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1. Dec 5, 2017

### proton4ik

1. The problem statement, all variables and given/known data
Hello!
I have some problems with constructing Lagrangian for the given system:

(Attached files)

2. Relevant equations
The answer should be given in the following form: L=T-U=...

3. The attempt at a solution
I tried to construct the Lagrangian, but I'm not sure if I did it correctly:
$$\frac{M}{2} \left(\dot z^{2} + \dot x^{2} + \dot y^{2}\right) = T$$
$$U=mgz$$
$$x=L*{\cos{\varphi}},$$
$$y=L*{\sin{\varphi}},$$
$$z=l*(1-{\cos{\varphi}})$$
$$- e l q \sin{\left (\varphi \right )} - g l M \left(- \cos{\left (\varphi \right )} + 1\right) + \frac{L^{2} M}{2} \left(\dot {\varphi}^{2} + \left(- \cos{\left (\varphi \right )} + 1\right)^{2}\right) = {\mathcal {L}}$$

I'll appreciate any help.

#### Attached Files:

• ###### 20171205_192712-1.jpg
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Last edited: Dec 5, 2017
2. Dec 5, 2017

3. Dec 6, 2017

### TSny

Please state the problem word for word (which should include a verbal description of the system).

4. Dec 6, 2017

### proton4ik

Construct Lagrangian and Lagrange's equations of the second kind for the given mechanical system. Are the equations integrable by quadratures? If yes, find the quadratures.

Unfortunately, there is no verbal description of the system.. (

5. Dec 6, 2017

### Staff: Mentor

Is there any description of the system? Equations? Something?

If not, I have no idea how anyone would answer the question.

6. Dec 6, 2017

### proton4ik

No, there's no description at all except for what's on the picture.

I suppose that it's a rod (L, M) suspended from two wires (l). There's probably a particle with charge q on the rod. And the system is rotating so the axis of rotation stays parallel to the wires.

7. Dec 6, 2017

### Staff: Mentor

Where is this problem from? I have never seen a textbook or other source give a picture with no accompanying words at all.

8. Dec 6, 2017

### Staff: Mentor

I would add that there appears to be a gravitational field pointing in the minus $z$ direction, and an electric field pointing in the plus $z$ direction. But I think all of this is tentative unless we can get more information about where this problem comes from.

9. Dec 6, 2017

### proton4ik

It's just an example of the exercise from the final exam. There are hundreds of similar pictures with the the same task in the examples.

10. Dec 6, 2017

### TSny

Since the problem is from an exam, it should not be too complicated. So, I tend to think that the system acts like a pendulum where rod $L$ always remains horizontal and parallel to the x axis. The two massless strings (or rods) $l$ remain parallel to each other and swing parallel to the y-z plane [EDIT: or maybe the system swings parallel to the x-z plane]. The angle $\phi$ is the angle that each string makes to the vertical.

I agree with Peter that there is a vertically upward uniform electric field and a massless point charge at the center of rod $L$.

This is just my guess at the intended interpretation.

Last edited: Dec 6, 2017
11. Dec 6, 2017

### Staff: Mentor

It's not a matter of complication; it's a matter of clarity. If that were all I saw on an exam, with no verbal description to tell me what the symbols mean, I would ask the examiner what they mean. (Unless the class in which this is an exam had provided a lot of context; perhaps the hundreds of similar pictures in the examples would help. But we can't see them here, so they're no help to us.)

12. Dec 6, 2017

### TSny

Absolutely
Right

13. Dec 6, 2017

### TSny

@proton4ik ,
You must have had your own interpretation of the problem in order to set up your equations in the first post. If you are interested in continuing with it, then let us know your interpretation and we might be able to help.

14. Dec 6, 2017

### proton4ik

Yes, of course, I already stated my interpretation :)
Here it is:
I suppose that it's a rod (L, M) suspended from two wires (l). There's probably a particle with charge q on the rod. And the system is rotating so the axis of rotation stays parallel to the wires.

15. Dec 6, 2017

### TSny

Bear with me, I can't visualize this. Would you mind describing the orientation and location of the axis of rotation a little more clearly? If the axis of rotation stays parallel to the wires, I take it that the wires stay parallel to one another. Does the axis of rotation itself change during the motion?

16. Dec 6, 2017

### proton4ik

I think the axis of rotation is shown on the picture (in the centre of the system). It doesn't change itself. And the angular velocity is $$\dot {\varphi}$$.
Yes, of course, I think that wires should stay parallel to one another.

17. Dec 6, 2017

### TSny

Does the entire system (including the upper supporting rod) rotate as a whole about the central vertical axis?

18. Dec 6, 2017

### proton4ik

Yes, I think so. I also suppose that it's moving back and forth so the z-coordinate is changing.

19. Dec 7, 2017

### TSny

Wouldn't any back and forth motion cause the wires to no longer be vertical (that is, the wires would not stay parallel to the axis of rotation)?

Also, is the angular velocity $\dot \varphi$ kept constant by an external agent, or is $\varphi$ a free variable (degree of freedom)?

20. Dec 7, 2017

### proton4ik

You're right. In this case, I guess there's only rotational motion. But how to find the potential energy of the system then?
$\varphi$ is probably a free variable.