# Lagrangian of a particle + conserved quantities

1. May 12, 2010

### fluidistic

1. The problem statement, all variables and given/known data
Consider the spherical pendulum. In other words a particle with mass m constrained to move over the surface of a sphere of radius R, under the gravitational acceleration $$\vec g$$.
1)Write the Lagrangian in spherical coordinates $$(r, \phi, \theta)$$ and write the cyclical coordinates and conserved quantities.
2)Define an effective potential energy $$U(\phi)$$ and determine (at least qualitatively) the allowed regions for the motion of the particle.
2. Relevant equations

Lots of.

3. The attempt at a solution
1)As the Lagrangian, I got $$L=\frac{m}{2}r^2 \left [ \dot \phi ^2 + \dot \theta ^2 \sin ^2 (\phi) \right ]+mgR \left [ 1- \cos (\theta) \right ]$$. Notice that $$\dot r=0$$.
I can already say that the energy is conserved since the Lagrangian doesn't depend explicitly on time.
I want to see if the linear momentum is conserved (is it stupid? I mean it seems obvious that no...)
So $$\vec P=\frac{\partial L}{\partial \vec \dot q}$$. For $$\dot q = \dot r$$, I get $$\vec P=\vec 0$$. What does that mean? It's a constant. So the "r" component of the linear momentum is conserved? It doesn't make sense to me to say it under these words.
I also reach $$\frac{\partial L}{\partial \dot \phi}=mR^2 \dot \phi$$. And $$\frac{\partial L}{\partial \dot \theta}=m \dot \theta \sin ^2 (\phi)$$.
Since the energy is conserved, I can apply Euler-Lagrange's equation. I get that $$\dot \phi =0$$ which makes sense to me. Thus $$\frac{\partial L}{\partial \dot \phi}=0$$ and so the $$\phi$$ component of the linear momentum is conserved. (Does this make sense to talk like this? What should I say instead? Maybe the angular momentum with respect to $$\phi$$?).
Lastly I found out that $$\frac{\partial L}{\partial \dot \theta}$$ is not necessarily constant (it is constant if the pendulum becomes the coplanar pendulum I believe) and thus the $$\theta$$ component of the linear momentum is not conserved.

I don't know what are the cyclical coordinates. I'll look into this further.
2)I just opened Landau's book to check out the effective potential energy and it is $$\frac{M^2}{2m R^2}$$ which seems to assume 2 particles... in my exercise there's only a massless sphere and a particle. I'm no clue what to do here.