Finite and infinitesimal Rotations

[Josh1079]

Hi,

I'm not sure about where I should post this question, so sorry in advance if I posted it in the wrong place.

My question is basically this screenshot. So I really have some difficulty in understanding the two equations. I mean how can it not be equal? I understand that rotations are non-commutative, but I really don't see why mathematically these two lines are not equal. Doesn't that violate the properties of matrix exponentials?

Thanks!

 

[TeethWhitener]

In general, if $A$ and $B$ are non-commuting operators, $e^{A+B} \neq e^A e^B$. You can see this most easily by expanding each side in a Taylor series. The general solution is known as the Baker-Campbell-Hausdorff formula.

EDIT: actually, probably the easiest way to see this is to note that $e^{A+B}=e^{B+A}$ but $e^A e^B \neq e^B e^A$ for noncommuting $A$ and $B$.

 

[Josh1079]

Ah...I see that now...didn't think it carefully enough

Thanks TeethWhitener!