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Need help solving a differential equation for orbit.

  1. Jun 23, 2014 #1
    I want to be able to map the position of a planet given initial position, velocity, and acceleration.

    I know the equation for Gravitational force (Newtonian) is: F=-GMm/r^2

    Using Newtons second law this gives: m(d^2x/dt^2)=-GMm/x^2

    Then simple Algebra yields: (d^2x/dt^2)+GM/x^2=0

    I understand you need initial conditions to solve this problem, so I'm going to say that
    Thank you very much to anyone who helps me out with this!
  2. jcsd
  3. Jun 23, 2014 #2


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    Gold Member

    Multiply the equation of motion by v=dx/dt and integrate.
    This will lead you to the conservation of energy:

    v²/2 - GM/x = Constant = b²/2 - GM/a

    Solve for v = dx/t, and integrate once more.
    (here you have a difficulty: there are two solutions)

    I assumed you were asked to solve the 1-dimensional problem, not the more realistic 2-d problem.
  4. Jul 30, 2014 #3
    If you want to solve the 2d problem, you should work whit polar coordinates.
    Also you are giving more initial conditions that you need.
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