Lagrangian of a system (First pages of L&L)

[fluidistic]

Homework Statement

The problem can be found in L&L's book "Mechanics" in the end of the first chapter. (See the last picture of page 12 of http://books.google.com.ar/books?id...v+Davidovich+Landau"&cd=2#v=onepage&q&f=false ). The mass m1 is free to move on the x-axis. While the mass m2 moves like a pendulum. There's the gravitational acceleration \vec g. I must find the Lagrangian of such a system.
I keep getting L=\frac{m_1 \dot x ^2}{2}+\frac{m_2 }{2} \left [ \dot x ^2 +2 \dot{\vec x} \dot \theta l \cos (\theta) + \dot \theta ^2 l^2 +2gl \cos (\theta) \right ]. I see that I have an error: I have a vector \dot {\vec x } which is impossible.
L&L's answer is L\frac{1}{2} (m_1+m_2) \dot x^2 +\frac{1}{2}m_2 (l^2 \theta ^2 +2l \dot x \theta \cos \theta)+m_2 g l \cos \theta.
I don't understand what I'm doing wrong. Also L&L don't have any \dot \theta term... Mine appeared when I calculated T_2, the kinetic energy of m_2 in polar coordinates. When I derivated the position \vec r_2 =(\dot {\vec x} + l \sin \theta )\hat i + (l \cos \theta) \hat j with respect to time I got some \dot \theta terms.

 

[Gokul43201]

Looks like a couple of typos in that version of L&L - the l \theta terms in the solution ought to be l \dot \theta. You can tell by inspection that the posted solution is dimensionally off (missing a factor of 1/time).

As for your concern about having an vector \mathbf{\dot x} ... what about the other vector in that term: \mathbf{\dot \theta}?

 

[fluidistic]

Looks like a couple of typos in that version of L&L - the l \theta terms in the solution ought to be l \dot \theta. You can tell by inspection that the posted solution is dimensionally off (missing a factor of 1/time).

As for your concern about having an vector \vec \dot x ... what about other vector in that term: \vec \dot \theta?

Ok thanks for the clarification.
If I understand you, \theta is a vector? I'm not really swallowing it. Can you confirm this? It would makes sense anyway, but I don't understand it. Hmm not sure it would make sense since I'd have a \cos (\text{vector}) term which doesn't seem OK.

 

[Gokul43201]

No, my LaTeX is messed up. For some reason, I can't get inline tex tags to work properly (with \vec and \dot). Fixing it ... give me a minute.

EDIT: Fixed now.

 

[fluidistic]

No, my LaTeX is messed up. For some reason, I can't get inline tex tags to work properly (with \vec and \dot). Fixing it ... give me a minute.

EDIT: Fixed now.

Ok perfect. I just found the same problem "explained in wikipedia" (see http://en.wikipedia.org/wiki/Lagrangian_mechanics#Pendulum_on_a_movable_support) but I still have my problem with \dot \vec x.
Here is what I do: \vec r_2 = (\vec x + l \sin \theta)\hat i +(l \cos \theta) \hat j.
In order to find the kinetic energy of m_2 (or m in wikipedia's article), I need the velocity squared. The velocity is \dot \vec r_2 =(\dot \vec x + \dot \theta l \cos \theta)\hat i +(- \dot \theta l \sin \theta) \hat j. Am I right until now?
So that |\dot \vec r_2|^2=(\dot \vec x + \dot \theta l \cos \theta)^2+(- \dot \theta l \sin \theta)^2= \dot x ^2 + 2 \underbrace{\dot \vec x}_{\text{this term}} \dot \theta l \cos( \theta ) + \dot \theta ^2 l^2 \cos ^2 (\theta).
EDIT: I FIND MY MISTAKE! A beginner one. Of course, there is no vector. I expressed the components of r (a vector) as vectors! Problem solved!
Thanks for your time. Now I go back to play with these Lagrangians. :)

 

[fluidistic]

Out of curiosity, I've plugged the Lagrangian into Euler-Lagrange equation and since I have 2 generalized coordinates (x and theta), I've found 2 equations. Namely the one that is in wikipedia: \ddot x \cos (\theta) +l \ddot \theta + g \sin (\theta)=0 if I take \theta as variable and another one if I take x as variable: m_2 \ddot x +2l \left [ \ddot \theta \cos (\theta) -\dot \theta ^2 \sin \theta \right ]=0.
Are they both the equations of motion? Or there's just one of these equations that describe the whole motion of the system?

 

[Gokul43201]

Are they both the equations of motion? Or there's just one of these equations that describe the whole motion of the system?

How many equations would you require to describe the motion of say a projectile?

 

[fluidistic]

How many equations would you require to describe the motion of say a projectile?

I think I made an error in my last post (I've redone the arithmetics and got a different answer), but my question doesn't change.
In this case I think I'd need 2 equations. One for the horizontal component of the position and one for the vertical component of the position vector; that is if I'm using Cartesian coordinates. If I derivate I get the velocity and if I derivate again I get the acceleration. I'd be done if I'm given the 2 initial conditions (\vec x_0, \vec v_0). I hope I'm right on this.

But still, in the case of this pendulum I get 2 equations involving theta, x and their derivatives. Both equations are worth 0. Hence I can equate them and write the first minus the second =0 and I've all the information within 1 single equation. Is this right?