System of ODEs in a rotating coord. system

[kostoglotov]

Homework Statement

imgur link: http://i.imgur.com/pb14Q4Q.png

Homework Equations

The Attempt at a Solution

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The thing I don't understand is where the first two terms of each 2nd order ODE came about.

I understand that they are there because the coordinate system is rotating, but when I set a rotating coord. system and try to get $x{''}_1 = x_1 + 2x{'}_2$ and $x{''}_2 = x_2 - 2x{'}_1$ I get $x{''}_1 = \alpha x{'}_2$ and $x{''}_2 = - \alpha x{'}_1$ where $\alpha$ is the constant angular velocity.

My reasoning is, let r = 1 (the distance between origin and any (x,y)), let A be the initial angle prior to some rotation and $\alpha t$ be the rotation rate by time.

$$x = \cos{A-\alpha t}$$

$$x{'} = \alpha \sin{A-\alpha t}$$

$$x{''} = -\alpha^2 \cos{A-\alpha t}$$

Do the same thing for y and wind up with $x{''}_1 = \alpha x{'}_2$ and $x{''}_2 = - \alpha x{'}_1$

That looks like it's on it's way to being $x{''}_1 = x_1 + 2x{'}_2$ and $x{''}_2 = x_2 - 2x{'}_1$...but I'm missing something.

 

[BvU]

Check out rotating frames of reference. One term is centrifugal, the other is Coriolis.