Particle in a central potential and a magnetic field: nearly circular orbits

[atat1tata]

Homework Statement

A particle of mass m and charge e moves in a central field with potential $V(r) = -\alpha r^{-3/2}$ and in a constant magnetic field $\vec{B} = B_0 \hat{e}_z$

1) Write the Lagrangian and the Hamiltonian

2) Write the first integral $H_{orb}$ for the equation of the orbit $u(\phi )$ ("u" refers to the inverse of the radius of the orbit), then calculate the radius $r_c$ for the stable circular orbit

3) For nearly circular orbit the equation of the orbit is $r = \frac{r_c}{1 + \epsilon \cos (\omega\phi)}$. Determine $\omega$ and $\epsilon$ for nearly circular orbits

Homework Equations

The equation my professor uses for the generalized potential generated by a magnetic field: $1/2 \vec{B}\cdot\vec{r}\times\vec{B}$

The Attempt at a Solution

1) I think I solved for the Lagrangian and the Hamiltonian. I'm using polar coordinates:
$L = 1/2m(\dot{r}^2 + r^2\dot{\phi}^2 + \alpha r^{-3/2} + 1/2 e B_0 r^2\dot{\phi}$

$H = \frac{p_r^2}{m} -\alpha r^{-3/2} + \frac{1}{2mr^2} (p_\phi - 1/2 e B_0 r^2)^2$

I had the impression that the equation my professor gave me for the magnetic generalized potential ($1/2 \vec{B}\cdot\vec{r}\times\vec{B}$) had its sign wrong, so I changed its sign. Am I correct?

2) Here the problems begin. I attempted splitting the Hamiltonian into a "kinetic part" and an effective potential ($V_{eff}(r) = -\alpha r^{-3/2} + \frac{1}{2mr^2} (p_\phi - 1/2 e B_0 r^2)^2$), then I tried to express the kinetic part as a function of $\frac{du}{d\phi}$ with several changes of variables. I ended up with $H = \frac{(p_{\phi}-1/2 e B_0/u^2)^2}{m}\frac{du}{d\phi} + V_{eff}(1/u)$. Something, however goes wrong, since i found a dependence on "u" (= 1/r) in the "kinetic energy part" of the Hamiltonian. I'm basically stuck here. I know that I should find the minima of the effective potential in order to find the circular orbits, but I find a very difficult equation to solve, so I'm asking if I am following the right path

3) For this question I thought to set up $V_{eff}(1/u_c) = \frac{(p_{\phi}-1/2 e B_0/u^2)^2}{m}\frac{du}{d\phi} + V_{eff}(1/u_c) + 1/2 V''_{eff} (1/u - 1/u_c)$ and then to try to solve it with separation of variables. For now I have not found $u_c$ so I don't know if it will work, so I am asking for confirmation

 

[Nate810]

or for a different approach.I apologize for the long post and for my poor English. I hope you can understand. Any help is much appreciated!A:1) The sign of the potential is correct.2) In order to find the radius of the circular orbit, you need to solve $\frac{dV_{eff}(r)}{dr}=0$, but you should use the equation for $V_{eff}(r)$ that you wrote in the attempt section.3) You should use the first integral to find the equation of the orbit. Then you can separate variables and the solution will be something like $r=\frac{r_c}{1+\epsilon \cos(\omega \phi)}$. The value of $\omega$ will be given by the first integral, while the value of $\epsilon$ will be determined by the initial conditions of the particle.