Results 1 to 5 of 5

Math Help - Solving Systems of Vector Differential Equations

  1. #1
    Newbie
    Joined
    Aug 2009
    Posts
    3

    Solving Systems of Vector Differential Equations

    Hello,

    I have a question regarding solving differential equations, in particular systems of differential equations, that involve vectors. For example, my particular problem is attached to this post (solution variables are alpha and Ac). While I'm no stranger to either vectors or solving ODEs, I'm not particularly sure how to go about solving a system like this when it involves cross -products, differentiation, and the like all at once. Does anyone have any tips on a solution method? I can handle either numerical or analytical solutions, I'm just rather confused by all the cross products in this problem.

    Thanks!
    Attached Thumbnails Attached Thumbnails Solving Systems of Vector Differential Equations-eqns.jpg  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member malaygoel's Avatar
    Joined
    May 2006
    From
    India
    Posts
    648
    Quote Originally Posted by Arrow View Post
    Hello,

    I have a question regarding solving differential equations, in particular systems of differential equations, that involve vectors. For example, my particular problem is attached to this post (solution variables are alpha and Ac). While I'm no stranger to either vectors or solving ODEs, I'm not particularly sure how to go about solving a system like this when it involves cross -products, differentiation, and the like all at once. Does anyone have any tips on a solution method? I can handle either numerical or analytical solutions, I'm just rather confused by all the cross products in this problem.

    Thanks!
    It seems you are rather confused by all the cross products in this problem.

    This may look the problem easier.
    \frac{d\overrightarrow{\alpha}}{dt}X(\frac{d\overr  ightarrow{\alpha}}{dt}X\overrightarrow{r})

    = \frac{d\overrightarrow{\alpha}}{dt}(\frac{d\overri  ghtarrow{\alpha}}{dt}\cdot \overrightarrow{r}) -\overrightarrow{r}(\frac{d\overrightarrow{\alpha}}  {dt}\cdot \frac{d\overrightarrow{\alpha}}{dt})

    Simply subtract the first and second equations to eliminate \alpha_c

    and then you will require some vector algebra formulas.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2009
    Posts
    3
    Thank you for the reply, malaygoel.

    I've taken a look at your suggestion. It was, in fact, one that had occurred to me in the past. Perhaps I'm missing a crucial step, but I could not see where to take the math at that point. What I appear to end up with is attached. My only thought at this point is to undistribute (as it were) the d(alpha)/dt DOT d(alpha)/dt terms, and then see if there's a useful identity for [A] DOT [A] that allows for further simplification. It doesn't seem to go anywhere helpful, though. Can you offer further suggestions?

    I should reiterate that my ultimate goal is to have a solution for Ac and a solution for alpha or d(alpha)/dt. All the other terms I've given (r1, r2, a1, a2) are considered "known".

    Thank you for the help again.
    Attached Thumbnails Attached Thumbnails Solving Systems of Vector Differential Equations-eqns2.jpg  
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Aug 2009
    Posts
    3
    So it turns out I realized I had set up the problem fundamentally wrong, which may explain why I was having issues. The problem I am attempting to solve is not the one I have in my first post and second post, but the equation attached to this post. My end goal is to end up with an equation in the form:

    omega_dot = ...

    I can then use this to numerically solve the ODE and move on. However, I'm still stuck with those cross products on the omega_dot term and I'm not sure how to get rid of them and end up with an equation I can deal with numerically. Are there any suggestions?

    Thank you.

    EDIT: If I take everything but "omega_dot X r1 - omega_dot X r2" to the other side, leaving "omega_dot X r1 - omega_dot X r2 = ...", I realize I can transform this to "omega_dot X (r1 - r2) = ...". If I can get rid of the (r1 - r2) term, I'm home free, though I'm currently not seeing it at the moment. Any advice?

    Put another way:
    \vec{a} \times \vec{b} = \vec{c}
    Solve for \vec{a}
    Attached Thumbnails Attached Thumbnails Solving Systems of Vector Differential Equations-eqn3.jpg  
    Last edited by Arrow; August 8th 2009 at 01:52 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member malaygoel's Avatar
    Joined
    May 2006
    From
    India
    Posts
    648
    Quote Originally Posted by Arrow View Post

    Put another way:
    \vec{a} \times \vec{b} = \vec{c}
    Solve for \vec{a}
    There is no unique solution

    \vec{a}=\lambda \vec{b}+\mu (\vec{b} \times \vec{c})

    where you can find out that
    \mu=\frac{1}{b^2}
    and \lambda \mbox{ is any arbitrary value}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: November 30th 2011, 01:41 AM
  2. Systems of linear Differential Equations
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: October 24th 2011, 02:05 AM
  3. Eigen Vector and solving ODE systems
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: December 12th 2009, 10:30 AM
  4. Systems of differential equations...
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 10th 2009, 06:36 AM
  5. Linear systems of differential equations
    Posted in the Calculus Forum
    Replies: 7
    Last Post: April 29th 2007, 11:45 AM

Search Tags


/mathhelpforum @mathhelpforum