Solving Systems of Vector Differential Equations

**Arrow** · August 7th 2009, 04:11 PM

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!

**malaygoel** · August 7th 2009, 08:07 PM

Originally Posted by Arrow

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.

**Arrow** · August 7th 2009, 08:59 PM

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.

**Arrow** · August 8th 2009, 10:46 AM

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 $\vec{a}$

**malaygoel** · August 8th 2009, 05:02 PM

Originally Posted by Arrow

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 $\lambda \mbox{ is any arbitrary value}$

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