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Full solution for the simple pendulum

  1. Sep 13, 2014 #1
    Having recently completed a session on the simple pendulum in physics, I was curious as to what the solution to θ''+(g/l)sin(θ)=0 for θ(t) was sans the sin(θ)=θ simplification.
  2. jcsd
  3. Sep 14, 2014 #2
    What have you done to satisfy your curiosity?
  4. Sep 14, 2014 #3
    Elliptic integral is the key word.
  5. Sep 14, 2014 #4
    voko, prior to posting I had essentially spent about an hour or so looking through various articles, lecture notes,et c., and hadn't run into anything involving a full solution of that ODE. I did check Wolfram Alpha which pointed me in the direction of the Jacobi elliptic functions, however that did not readily lend itself to understanding the derivation, or finding a numerical solution. I looked under functions.wolfram.com for the Jacobi Amplitude function, and wasn't able to find what I was looking for
  6. Sep 14, 2014 #5
    The full solution of that ODE is not expressible in elementary functions. Have a look here: http://en.wikipedia.org/wiki/Pendulum_(mathematics [Broken])
    Last edited by a moderator: May 6, 2017
  7. Sep 15, 2014 #6
    Multiply [tex]\theta^.[/tex] to make [tex]{{\theta^.}^2}^. [/tex], integrate and then take square root to get [tex]\frac{dt}{d\theta}[/tex]
  8. Sep 15, 2014 #7
    Or just start with conservation of energy and get the first-order ODE directly.
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