Constrained mass point subject to elastic force and weight

[Ocirne94]

Hi all, is my solution correct? I was rejected because of this...

Homework Statement

Consider a mass point (mass = m) constrained to move on the surface of a sphere (radius = r). The point is subject to its own weight's force and to the elastic force of a spring (elastic constant = k, rest length = 0) which at the other end is fixed to the sphere's north pole.

Write the Lagrangian and the Hamiltonian of the system.
Write the Lagrange's and Hamilton's equations of motion.
Find the constants of motion.
Give a qualitative description of the point's movement

Homework Equations

None given.

The Attempt at a Solution

There are 2 degrees of freedom. I choose spherical coordinates theta and phi [but is this correct? The point can reach the poles, where those coordinates aren't defined anymore].
The kinetic energy is
$T = \frac{m}{2}\cdot (r^2 \dot\theta^{2} + r^{2}sin(\theta)^{2}\dot\phi^{2})$
The potential energy is
$V = \kappa\cdot r^{2} (1-cos\theta) + mgr(1+cos\theta)$
The Lagrangian is simply
$L = T-V$
and, since there isn't any explicit dependence on time, the Hamiltonian is simply
$H = T+V$, but expressed as a function of the momenta $p_\theta$ and $p_\phi$. I computed it as $p_\theta \cdot \dot\theta + p_\phi \cdot \dot\phi - L$

$p_\theta = \frac{\partial L}{\partial \dot\theta}=mr^2\dot\theta$
$p_\phi = \frac{\partial L}{\partial \dot\phi}=\dot\phi r^{2} sin(\theta)^{2}m$

Then Lagrange's equations are only computations (I hope I haven't mistaken the derivatives), and so are the Hamilton's.

$p_\phi$ is a constant of motion; the total energy (H or E) is, too. There aren't other constants of motion.

Then I have drawn the chart of V and I have used it to trace a qualitative phase portrait, and I have made basic observations on equilibrium points (one, unstable, when the point is at the south pole; one, stable, when it is at the north pole; and a circumference (a parallel) depending on the mass and the elastic constant.

And now?

Thank you in advance
Ocirne

 

[haruspex]

The potential energy is
$V = \kappa\cdot r^{2} (1-cos\theta) + mgr(1+cos\theta)$

How do you get (1-cos(θ))? And shouldn't there be a factor 1/2 on that term?

 

[Ocirne94]

I get it from geometry: the spring's square length is

$r^2(1-cos\theta)^{2} + (sin\theta^{2})$

which becomes

$r^2 + r^2cos\theta^2-2r^2cos\theta+r^2sin\theta^2$

the 2 gets simplified with the 1/2 of the elastic potential formula.
This (I forgot to say) setting potential=0 at the south pole of the sphere.

 

[haruspex]

I get it from geometry: the spring's square length is

$r^2(1-cos\theta)^{2} + (sin\theta^{2})$

Ah yes, of course.
Everything else looks reasonable to me. Maybe more is wanted on the qualitative description. In general, it will oscillate above and below a latitude corresponding to a stable horizontal orbit, yes? Might it be SHM, in terms of a suitable function of time?

 

[HalfLostAndSearching]

Hello Ocirne, thank you for sharing your solution with us. Your approach seems to be generally correct, however there are a few points that could be improved upon. Firstly, your choice of spherical coordinates is appropriate for this problem since the point is constrained to the surface of a sphere. However, as you mentioned, at the poles the coordinates are not well defined. In this case, it is better to use a different set of coordinates such as latitude and longitude, or spherical coordinates with a different choice of axis. This will ensure that your equations are well-defined throughout the entire motion of the point.

Secondly, your expression for the potential energy seems to be missing a term. The potential energy should also include the energy of the spring, which is given by 1/2*k*(r-r0)^2, where r0 is the rest length of the spring. This term will contribute to the equations of motion and will also affect the equilibrium points of the system.

Thirdly, you are correct in saying that p_phi is a constant of motion, but there is also another constant of motion in this system which is the z-component of the angular momentum, Lz. This can be derived from the symmetry of the problem and will also affect the equations of motion and the equilibrium points.

Finally, your qualitative description of the point's movement is a good start, but you could also mention the effect of changing the mass and the elastic constant on the motion. For example, a higher mass or a stiffer spring will result in a smaller range of motion for the point.

Overall, your solution is on the right track, but there are a few areas that could be improved upon. Keep up the good work!