Results 1 to 2 of 2

Thread: How to iterate?

  1. #1
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    11,344
    Thanks
    828
    Awards
    1

    How to iterate?

    I have a non-linear 1st order differential equation and I would like to find a way to iterate it. I thought I knew how to approach it but I've hit a stumbling block. Here's the equation:
    $\displaystyle \dfrac{d \theta}{dt} = \sqrt{ \dfrac{2g}{R} } \sqrt{ sin( \theta ) - sin( \theta _0 ) }$

    Where $\displaystyle \theta _0$ is taken to be the angle at time 0. (This is from the energy equation for the simple pendulum. For the usual simple pendulum we take $\displaystyle \theta$ small. We don't have that here...it's the general case. I can provide the derivation if someone likes.)

    Now, I can iterate this in $\displaystyle \theta$ :
    $\displaystyle \dfrac{ \theta _{n + 1} - \theta _n }{\Delta t} = \sqrt{ \dfrac{2g}{R} } \sqrt{ sin( \theta _n ) - sin( \theta _0 ) }$

    becomes
    $\displaystyle \theta _{n + 1} = \theta _n + \sqrt{ \dfrac{2g}{R} } \sqrt{ sin( \theta _n ) - sin( \theta _0 ) } \cdot \Delta t$
    where $\displaystyle \Delta t$ is suitably small and taken to be constant. ( $\displaystyle sin( \theta _n ) \geq sin( \theta _0 )$ for any physical case.)

    However, what I'm really after is the time series so I can investigate the period of the motion and such. Here's where the problem lies. Similar to before:
    $\displaystyle \dfrac{ \Delta \theta }{ t_{n + 1} - t_n} = \sqrt{ \dfrac{2g}{R} } \sqrt{ sin( \theta _n) - sin( \theta _0 ) }$

    becomes
    $\displaystyle t_{n +1} = t_n + \sqrt{ \dfrac{R}{2g}} \cdot \dfrac{1}{\sqrt{ sin( \theta _n ) - sin( \theta _0 ) }} \cdot \Delta \theta$
    Where $\displaystyle \Delta \theta $ is suitably small and taken to be constant.

    Sounds good, right? Everywhere except the highest points of the motion where $\displaystyle sin( \theta _n ) = sin( \theta _0)$. Then it becomes... "non-trivial." I'm sure the differential equation is right so there must be an error in my idea of how to iterate this.

    How do I approach this?

    Thanks!

    -Dan
    Last edited by topsquark; Nov 16th 2018 at 08:08 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    11,344
    Thanks
    828
    Awards
    1

    Re: How to iterate?

    (Ahem!) I just took a little time to work out the angle series and that one's hosed, too.

    Could someone be kind enough to show me how I might go about iterating either or both of these? Looks like I know less about this than I thought.

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. functions - Iterate and Curry
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Sep 14th 2011, 09:14 AM

/mathhelpforum @mathhelpforum