Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Solving this differential equation

  1. Jul 6, 2005 #1
    Hello,

    while working on a simple mechanics problem using polar coordinates, I got this equation

    second derivative of phi = (-g/R) sin phi

    now I need to solve this to get an equation for phi (t) but the books says that I cannot solve this using elementary functions and that the solution will be the more complex Jacobi elliptic function. My question is why can't I integrate twice to get the equation for phi (t) ?

    thanks a lot
     
  2. jcsd
  3. Jul 6, 2005 #2
    You can't just straight integrate that differential equation. I'm not even sure how one would attempt it. Here's one way to set up the elliptic integral:

    Multiply through by [tex] \dot{\phi} [/tex] and get the differential equation
    [tex]\ddot{\phi} \dot{\phi} = -g/R \dot{\phi} sin \phi [/tex]
    which we recognize as being the first time derivative of
    [tex]\dot{\phi}^2 = g/R cos \phi + C [/tex]
    when then leads to the integral:
    [tex]\int d\phi ~ 1/\sqrt{g/R cos \phi + C} = t - t_0[/tex]
    which is then the elliptic integral left behind. This is why we make things like the small angle approximation, where applicable. If you're working with exact numbers, you could pretty simply write a numerical algorithm
     
  4. Jul 6, 2005 #3
    MalleusScientiarum ,

    why can't we straight integrate it with respect to time ?
     
  5. Jul 6, 2005 #4
    ok never mind....stupid question. i know why :)
     
  6. Jul 7, 2005 #5

    James R

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Yeah - because it's sine of phi, not sine of t.
     
  7. Jul 7, 2005 #6

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    In post #2, you missed a little "2" when going from [itex] \dot{\phi}\ddot{\phi} [/itex] to the square of the first derivative.

    Daniel.
     
  8. Jul 10, 2005 #7
    i guess theylor expansion is the best choice
     
  9. Jul 10, 2005 #8
    I advise numerical analysis.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Solving this differential equation
Loading...