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

1. Feb 9, 2012

### atat1tata

1. The problem statement, all variables and given/known data
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

2. Relevant equations

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

3. 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 dependance 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