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Homework Statement
Hi everybody! Here is a new Lagrange problem I am trying to solve, and I would like to have your opinion about my solution so far!
A barbell composed of two masses ##m_1## and ##m_2##, idealised as particles and separated by a distance ##a## from each other, moves in the Earth's gravitational field.
a) Write the Lagrange function using spherical coordinates for the relative vector and cartesian coordinates for the center of mass.
b) Derive the equations of motion and give minimum one special solution.
c) Which quantities are conserved in this system?
Homework Equations
Lagrange function, Lagrange equations of motion
The Attempt at a Solution
So first I drew the "situation" (see attached picture) and forgot everything about a barbell. I assumed ##m_2## was bigger than ##m_1## to have an idea of what's going on, but that' not so important. I called ##\vec{R}## the vector going from the origin to the center of mass and ##\vec{r}## the vector going from ##m_2## to ##m_1##. ##\vec{r_1}## and ##\vec{r_2}## are the vectors going from the origin to respectively ##m_1## and ##m_2##.
a)
So first I wrote ##\vec{R}## and ##\vec{r}## in terms of ##\vec{r_1}## and ##\vec{r_2}## :
##\vec{R} = \frac{\vec{r_1} \cdot m_1 + \vec{r_2} \cdot m_2}{m_1 + m_2}##
##\vec{r} = \vec{r_1}  \vec{r_2}##
I define the reduced mass as ##m := \frac{m_1 \cdot m_2}{m_1 + m_2}## and I rewrite those to get new expressions for ##\vec{r_1}## and ##\vec{r_2}##:
##\vec{r_1} = \frac{m}{m_1} \vec{r} + \vec{R}##
##\vec{r_2} = \frac{m}{m_2} \vec{r} + \vec{R}##
Preliminary, I wrote ##T## and ##V##:
##T = \frac{1}{2} m_1 \cdot \dot{r}_1^2 + \frac{1}{2} m_1 \cdot \dot{r}_2^2##
##V = m \cdot g \cdot z##
Of course ##m## is still the reduced mass and so ##z## refers to the zposition of the center of mass. After substituting ##\vec{r_1}## and ##\vec{r_2}##, I get an expression for ##L##:
##L = \frac{1}{2} m \cdot \dot{\vec{r}}^2 + \frac{1}{2} (m_1 + m_2) \dot{\vec{R}}^2  m \cdot g \cdot z##
That's nice. But is it correct? Unfortunately the problem asks for ##\vec{R}## to be expressed in cartesian coordinates and ##\vec{r}## to be expressed in spherical coordinates... Therefore:
##\vec{R}^2 = x^2 + y^2 + z^2 \implies \dot{R}^2 = \dot{x}^2 + \dot{y}^2 + \dot{z}^2##
##\vec{r} = r \cdot (\cos \theta, \sin \theta) \implies \dot{\vec{r}} = \dot{r} \cdot (\cos \theta, \sin \theta) + r \cdot \dot{\theta} \cdot (\sin \theta, \cos \theta)## since ##\varphi## is constant (I would think) because both masses accelerate at the same rate ##g##.
And that brings me to that Lagrange function:
##L = \frac{1}{2} (m_1 + m_2) (\dot{x}^2 + \dot{y}^2 + \dot{z}^2) + \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2)  mgz##
Uuf...Is that correct? I'm not completely sure about the transformation from vectors to scalars, especially in the case of ##\vec{r}##; I've squared "inside" because I think the kinetic energy should square the velocity in direction of ##\hat{r}## and also in direction of ##\hat{\theta}##. Is that right?
b)
So hopefully a) is correct to begin with. I'm a bit confused by the next question because if it was up to me I would have written the Lagrange function as well as the equations of motion in terms of ##R## and ##r##, but because of the choice of coordinates in the end I had to write them in terms of ##z## and of ##r## (because the other equations give a ##0## acceleration, I will write them anyway). I get those:
##\ddot{x} = 0##
##\ddot{y} = 0##
##\ddot{\theta} = 0##
##\ddot{\varphi} = 0##
##\frac{\partial L}{\partial z} = m \cdot g## and ##\frac{d}{dt} \frac{\partial L}{\partial \dot{z}} = (m_1 + m_2) \cdot \ddot{z}##
##\implies \ddot{z} = \frac{m}{m_1 + m_2} g = \frac{ m_1 \cdot m_2}{(m_1 + m_2)^2} g##
##\frac{\partial L}{\partial r} = m \cdot r \cdot \dot{\theta}^2## and ##\frac{d}{dt} \frac{\partial L}{\partial \dot{r}} = m \cdot \ddot{r}##
##\implies \ddot{r} = r \dot{\theta}^2##
Mm... I don't know what to think of those. The square on ##m_1 + m_2## in the ##\ddot{z}## equation looks strange, and I have nothing to say really about the second one. At least the zeros in the other equations are coherent!
I didn't go further yet as I am too unsure about my solution. Can anybody take a look and tell me if that's correct/wrong?
Thanks a lot in advance for your answers!
Julien.
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