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2D Phase portrait - Black hole?

  1. Jun 8, 2015 #1
    1. The problem statement, all variables and given/known data

    Trajectories around a black hole can be described by ## \frac{d^2u}{d\theta^2} + u = \alpha \epsilon u^2 ##, where ##u = \frac{1}{r}## and ##\theta## is azimuthal angle.

    (a) By using ##v = \frac{du}{d\theta}##, reduce system to 2D and find fixed points and their stability. Find direction of fastest perturbations.
    (b) Sketch the phase portrait. Would stability of fixed points differ in the non-linearized version?

    2. Relevant equations


    3. The attempt at a solution

    Part (a)
    The equations now become ##\delta v = \delta \dot u## and ##\delta \dot v + \delta u = 2\epsilon u \delta u##.
    2010_B1_Q1.png
    Fixed points are ##\left( 0,0 \right)## and ##\left( \frac{1}{\epsilon}, 0 \right)##. At ##(0,0)##, all eigenvalues are imaginary, so the fixed point is a center. At ##(\frac{1}{\epsilon},0)##, eigenvalues are ##\pm 1## so fixed point is a saddle.
    Eigenvalue of ##J + J^T## is ##2\epsilon u## and direction of fastest perturbation is ##u=v##.

    Part(b)
    Eigenvalue in general is ##\lambda^2 = (2\epsilon u - 1)##, so for ##|u| > \frac{1}{2\epsilon}##, the particle doesn't get trapped by the black hole?

    2010_B1_Q1_2.png
     
  2. jcsd
  3. Jun 10, 2015 #2
  4. Jun 15, 2015 #3
    bumpp
     
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