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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
    x(0)=a
    x'(0)=b
    x''(0)=c
    Thank you very much to anyone who helps me out with this!
     
  2. jcsd
  3. Jun 23, 2014 #2

    maajdl

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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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