Lagrangian and lagrange equations of a system of two masses

[Dixanadu]

Homework Statement

Hi guys.

http://img189.imageshack.us/img189/5123/systemn.jpg

The image shows the situation. A pointlike particle of mass m is free to move without friction along a horizontal line. It is connected to a spring of constant k, which is connected to the origin O. A rigid massless rod joins the particle with another one of the same mass, which can oscillate freely as a pendulum. Gravity acts vertically downwards.

Here are the questions:

(1) How many degrees of freedom are there? Find appropriate generalized coordinates.
(2) Find the Lagrangian and the Euler-Lagrange equations.

There are some more but if i get the correct lagrange equations, I'm pretty sure I can do those.

Homework Equations

L = T - V; the Lagrangian
\frac{d}{dt}\frac{∂L}{∂\dot{q}}=\frac{∂L}{∂q} ; the Euler-Lagrange equation.

The Attempt at a Solution

Here are my answers, but I am not 100% sure if they are right:

(1) There are 2 degrees of freedom - the position of the mass connected to the spring, s, along the horizontal, and the angle between the vertical and the rod as it oscillates: θ.

(2) Here's a breakdown of what I've done. I reached the Lagrangian, I know how to get the Euler-Lagrange equation as well. But the problem is this - I end up with an equation that has s, \dot{s}, θ, \dot{θ}. So when I try to find the Euler-Lagrange equation, I end up with two sets of them. Is it meant to be that way?
http://img171.imageshack.us/img171/592/answerh.jpg

Thanks a lot guys! sorry for the super long post, but I wanted to give all the information I have.

 

[TSny]

Overall, looks pretty good to me except shouldn't the y coordinate of the pendulum bob be negative?
[Edit: Also, maybe a missing factor of $l$ in the middle term of $T$ and $T_{tot}$]

 

[Dixanadu]

Okay, thanks for the reply, I'll go through it again for the l factor. And yea I think the y coordinate should be negative :P failed to spot that.

But am I meant to end up with two Lagrange equations if I apply the derivatives to this?

 

[TSny]

Yes, one for each degree of freedom.

 

[Dixanadu]

heh, I guess that was a stupid question :P I'm gona work on the remaining parts of the question, but I think i'll need some help, so I'll be back. Thanks a bunch TSny!

 

[Dixanadu]

http://img835.imageshack.us/img835/1905/answer2.jpg

Here's what I get for the Euler-Lagrange equations. I have two problems now - firstly, is this even correct? i don't know why i get the feeling it isnt.

Secondly, the next part of the question is "Determine the position of stable and unstable equilibrium of the system, hence write down the Lagrangian describing the small oscillations about the stable equilibrium position. Determine the corresponding frequency of small oscillations."

No idea how to do that. Help please? :) thanks!

 

[TSny]

Looks good except for that factor of m in the first term of the last equation.

You can use the Euler-Lagrange equations of motion to determine the equilibrium points. What quantities in the equations would necessarily be zero if the system is sitting in an equilibrium position? After letting those quantities be zero, what do the equations of motion tell you about the values of s and θ?

 

[Dixanadu]

This is where I've gotten so far. I realize that you need to differentiate the total potential energy of the system with respect to each coordinate, (s, θ) and set this equal to 0. Here's what I've found. The problem is - what do i do next to find the frequency of small oscillations? Do i need the secular equation and all that long stuff :S
http://img853.imageshack.us/img853/5200/answer3.jpg

 

[TSny]

Looks very good. You can now pick which is stable and which is unstable. (Not much of a surprise here .)

 

[Dixanadu]

Okay so I'm being asked to write the small oscillation Lagrangian, which I have done in matrix form. Then of course I need to find the frequency of these oscillations about the stable equilibrium position which is obviously when s = θ = 0. I'm using the secular equation, and I end up with the determinant at the bottom.

The problem is...I can't solve for ω in that determinant...I need help again T_T

The equation just gets too long and I don't know what to do with it at all.

http://img138.imageshack.us/img138/6691/answer4e.jpg

 

[TSny]

The problem is...I can't solve for ω in that determinant...I need help again T_T

The equation just gets too long and I don't know what to do with it at all.

It should not be too bad. You might find it helpful to let $\lambda = \omega^2$ and get an equation for $\lambda$.