Results 1 to 2 of 2

Math Help - Getting around interacting sequences

  1. #1
    Newbie
    Joined
    Aug 2008
    Posts
    5

    Getting around interacting sequences

    The problem I'm currently facing is how to produce and expression to describe the path of a particle in order that it's position can be calculated with a given time. So similar to your suvat equations of motion, this is all fine and good until you start trying to derrive equations where the next x coordinate, for example, is dependent somehow on the last y an . The same is true for y. One instance of this is gravity between two bodies, the displacement on x and y of a body is dependent on the distance between the two, which sure enough can only be calculated from the previous x,y displacement and if i try to rearrange this I end up with an infinite cycle of substitution. Mandelbrot and attractors springs to mind.

    This seems to shows up everywhere in physics and I've not yet got my head round it. Effectively you have two interacting sequences, I've diluted this idea into a simple sequence where p and q are constants and a and b begin at 0:

    <br />
a_{n+1} = a_{n} * b_{n} +p<br />
    <br />
b_{n+1} = 2*a_{n} * (b_{n} - 3) +q<br />

    I've graphed this in quickly flash, excuse the scaleless graph! Fork it and add scales if you into as3 flash on 2010-5-19 | wonderfl build flash online
    so if you follow that link you'll see a spiral slowly changing before tightening up and apparently exploding. What happening is I'm iterating over that sequence 500 times and plotting each point (a = x, b=y), then once it's been plotted it's displayed, q is increased by 0.001 and it is calculated and replotted. This is happening 30 times a second so you can see a gradual change. Is this an attractor? I'm certainly new to all this stuff


    So my question is how can work around this? Say I wanted to derive and expression for the x & y of an paritcle if its is being attracted by a static body? Is this possible without iterating over each value?

    Thanks, apologies for the long post!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Geo877 View Post
    The problem I'm currently facing is how to produce and expression to describe the path of a particle in order that it's position can be calculated with a given time. So similar to your suvat equations of motion, this is all fine and good until you start trying to derrive equations where the next x coordinate, for example, is dependent somehow on the last y an . The same is true for y. One instance of this is gravity between two bodies, the displacement on x and y of a body is dependent on the distance between the two, which sure enough can only be calculated from the previous x,y displacement and if i try to rearrange this I end up with an infinite cycle of substitution. Mandelbrot and attractors springs to mind.

    This seems to shows up everywhere in physics and I've not yet got my head round it. Effectively you have two interacting sequences, I've diluted this idea into a simple sequence where p and q are constants and a and b begin at 0:

    <br />
a_{n+1} = a_{n} * b_{n} +p<br />
    <br />
b_{n+1} = 2*a_{n} * (b_{n} - 3) +q<br />

    I've graphed this in quickly flash, excuse the scaleless graph! Fork it and add scales if you into as3 flash on 2010-5-19 | wonderfl build flash online
    so if you follow that link you'll see a spiral slowly changing before tightening up and apparently exploding. What happening is I'm iterating over that sequence 500 times and plotting each point (a = x, b=y), then once it's been plotted it's displayed, q is increased by 0.001 and it is calculated and replotted. This is happening 30 times a second so you can see a gradual change. Is this an attractor? I'm certainly new to all this stuff


    So my question is how can work around this? Say I wanted to derive and expression for the x & y of an paritcle if its is being attracted by a static body? Is this possible without iterating over each value?

    Thanks, apologies for the long post!
    With a bit of manipulation you can find a second order non-linear recurrence for a_n, if my algebra is error free it is:

    a_{n+1}=2a_n^2-6a_na_{n-1}+a_a(q-2p)+p

    I am not familiar with solving such recurrences, but there may be a method of doing so.

    CB
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Microstates of N non-interacting Dipoles
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: February 12th 2011, 02:41 AM
  2. Convergence in sequences of sequences
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: October 19th 2010, 07:28 AM
  3. Sequences and the sequences' arithmetics
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 6th 2010, 09:31 PM
  4. Monotone sequences and Cauchy sequences
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 21st 2009, 08:59 PM
  5. Replies: 5
    Last Post: January 16th 2008, 04:51 PM

Search Tags


/mathhelpforum @mathhelpforum