Lagrangian of a particle + conserved quantities

[fluidistic]

Homework Statement

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 [tex]\vec g[/tex].
1)Write the Lagrangian in spherical coordinates [tex](r, \phi, \theta)[/tex] and write the cyclical coordinates and conserved quantities.
2)Define an effective potential energy [tex]U(\phi)[/tex] and determine (at least qualitatively) the allowed regions for the motion of the particle.

Homework Equations

Lots of.

The Attempt at a Solution

1)As the Lagrangian, I got [tex]L=\frac{m}{2}r^2 \left [ \dot \phi ^2 + \dot \theta ^2 \sin ^2 (\phi) \right ]+mgR \left [ 1- \cos (\theta) \right ][/tex]. Notice that [tex]\dot r=0[/tex].
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 [tex]\vec P=\frac{\partial L}{\partial \vec \dot q}[/tex]. For [tex]\dot q = \dot r[/tex], I get [tex]\vec P=\vec 0[/tex]. 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 [tex]\frac{\partial L}{\partial \dot \phi}=mR^2 \dot \phi[/tex]. And [tex]\frac{\partial L}{\partial \dot \theta}=m \dot \theta \sin ^2 (\phi)[/tex].
Since the energy is conserved, I can apply Euler-Lagrange's equation. I get that [tex]\dot \phi =0[/tex] which makes sense to me. Thus [tex]\frac{\partial L}{\partial \dot \phi}=0[/tex] and so the [tex]\phi[/tex] 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 [tex]\phi[/tex]?).
Lastly I found out that [tex]\frac{\partial L}{\partial \dot \theta}[/tex] is not necessarily constant (it is constant if the pendulum becomes the coplanar pendulum I believe) and thus the [tex]\theta[/tex] 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 [tex]\frac{M^2}{2m R^2}[/tex] 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.

 

[mark726]

Thank you for bringing up this interesting problem. I am always excited to discuss and analyze different physical systems. Let me address your questions and provide some insights on the spherical pendulum.

1) Your Lagrangian seems to be correct, good job! And yes, the energy is conserved since the Lagrangian does not depend explicitly on time. As for the linear momentum, it is not conserved in this system since there is a force acting on the particle (gravity) and the Lagrangian is not invariant under translations in space. However, the angular momentum with respect to the z-axis (perpendicular to the plane of motion) is conserved. This is because the Lagrangian is invariant under rotations around this axis.

The conserved quantities in this system are the total energy (as you mentioned), the angular momentum with respect to the z-axis, and the z-component of the angular momentum. The cyclical coordinates are those that do not appear explicitly in the Lagrangian, so they are r and \theta in this case. This means that their corresponding momenta, \dot r and \dot \theta, are conserved quantities.

2) The effective potential energy is a useful concept in analyzing the motion of a particle in a central force field. In this case, it represents the potential energy experienced by the particle as if it were moving in one dimension (along the \phi coordinate), with the same equations of motion. This effective potential energy is given by U_{eff}(\phi) = mgR(1-\cos\theta), where \theta is a function of time. This potential has a minimum at \theta = 0 (when the particle is at the bottom of the sphere) and a maximum at \theta = \pi (when the particle is at the top of the sphere). So, qualitatively, the allowed regions of motion for the particle are between the bottom and top of the sphere. The particle cannot go beyond these points as it would require infinite energy.

I hope this helps in your understanding of the spherical pendulum. Keep up the good work in your studies and keep exploring the fascinating world of physics!