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

  1. Feb 9, 2012 #1
    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 [itex]V(r) = -\alpha r^{-3/2}[/itex] and in a constant magnetic field [itex] \vec{B} = B_0 \hat{e}_z[/itex]

    1) Write the Lagrangian and the Hamiltonian

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

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


    2. Relevant equations

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

    3. The attempt at a solution

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

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

    I had the impression that the equation my professor gave me for the magnetic generalized potential ([itex]1/2 \vec{B}\cdot\vec{r}\times\vec{B}[/itex]) 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 ([itex]V_{eff}(r) = -\alpha r^{-3/2} + \frac{1}{2mr^2} (p_\phi - 1/2 e B_0 r^2)^2[/itex]), then I tried to express the kinetic part as a function of [itex]\frac{du}{d\phi}[/itex] with several changes of variables. I ended up with [itex]H = \frac{(p_{\phi}-1/2 e B_0/u^2)^2}{m}\frac{du}{d\phi} + V_{eff}(1/u)[/itex]. 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 [itex] 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)[/itex] and then to try to solve it with separation of variables. For now I have not found [itex]u_c[/itex] so I don't know if it will work, so I am asking for confirmation
     
  2. jcsd
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