DISCLAIMER: I would like to post a few things about what I am studying. Hopefully by writing all this I will get a better idea of it. There will probably be a few mistakes. Please feel free to comment on anything I said or add your own point of view, I'd love to hear it!(adsbygoogle = window.adsbygoogle || []).push({});

I have three vector fields in [itex]\mathbb{R}^3[/itex] given by

[tex]U=y\partial_z - z\partial_y[/tex]

[tex]V=z\partial_x - x\partial_z[/tex]

[tex]W=x\partial_y - y\partial_x[/tex]

I wish to find the local one-parameter group generated (or flow if you prefer) by [itex]U,V,W[/itex].

Now I thought that you could do this by solving an ODE for each vector field. For example, for U, we should have a coupled ODE...

[tex]\frac{dy}{dt} = -z[/tex]

[tex]\frac{dz}{dt} = y[/tex]

Similarly for V, we have

[tex]\frac{dz}{dt} = -x[/tex]

[tex]\frac{dx}{dt} = z[/tex]

and for W we have

[tex]\frac{dx}{dt} = -y[/tex]

[tex]\frac{dy}{dt} = x[/tex]

So we have three coupled first order linear differential equations.

Assuming we have the following initial conditions,

[tex]x(0) = A, y(0) = B, and z(0) = C[/tex]

the general solutions are (for U)

[tex]z(t) = C\cos t - B\sin t[/tex]

[tex]y(t) = C\sin t + B\cos t[/tex]

and it is easy to see what they are for V and W.

Now this is all easy stuff to do (assuming I got it right! :) ), it is basic ODE stuff. But I want to relate all this to the rotation group...

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# Homework Help: Help with Rotations

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