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1D three-body problem (with spherical shell)

  1. Aug 11, 2010 #1
    Let's suppose that we have two point particles with masses m1,m2 and the spherical shell with mass M, placed in a line, at distances h1,h2 and H from 0 in that line (0 is the center of some inertial frame of reference). The initial conditions and the equations of motion are the following:

    [itex]h_1(0)=H(0)=h_0[/itex]

    [itex]h_2(0)=0[/itex]

    [itex]h_1'(0)=h_2'(0)=H'(0)=0[/itex] (time derivative)

    (the mass m1 is in the center of the spherical shell)


    [itex]\frac{dh_1^2}{dt^2}=-G\frac{m_2}{(h_1-h_2)^2}[/itex]

    [itex]\frac{dh_2^2}{dt^2}=G\frac{m_1}{(h_1-h_2)^2}+G\frac{M}{(H-h_2)^2}[/itex]

    [itex]\frac{dH^2}{dt^2}=-G\frac{m_2}{(H-h_2)^2}[/itex]

    Is there any way to solve this problem (to find the positions of the masses as functions ot the time)?
     
  2. jcsd
  3. Aug 11, 2010 #2

    Pythagorean

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    Gold Member

    integrate twice with respect to t?

    Or are H and h actually arbitrary functions of t?
     
  4. Aug 11, 2010 #3
    Yes they are functions of t, i think, so it's more complicated.

    Edit: I'm going to create the same thread in Classical Physics section so the moderators can close this one. Thnx!
     
    Last edited: Aug 11, 2010
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