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1. Homework Statement
attached:
2. Homework Equations
where ##J_{yz} ## is
3. The Attempt at a Solution
In a previous question have exponentiated the generator ##J_{yz}## to show it is the generator of rotation around the ##x## axis via trig expansions
so ##\Phi(t,x,y,z) \to \Phi(t,x,y cos \alpha  z sin \alpha, y sin \alpha + z cos \alpha ) ## and so via small angle expansions have:
##y \to ( y(1\frac{\alpha^2}{2}+O(\alpha^4))z(\alpha\frac{\alpha^3}{3!}+O(\alpha^5))) ##
and
## z\to ( ( z(1\frac{\alpha^2}{2}+O(\alpha^4))+y(\alpha\frac{\alpha^3}{3!}+O(\alpha^5)))##
I am unsure now how expand. I thought perhaps a taylor expansion in multivariables  y and z was the idea, but I can't see how you would arrive at the answer attached with this:
Any tips appreciated.ta.
attached:
2. Homework Equations
where ##J_{yz} ## is
3. The Attempt at a Solution
In a previous question have exponentiated the generator ##J_{yz}## to show it is the generator of rotation around the ##x## axis via trig expansions
so ##\Phi(t,x,y,z) \to \Phi(t,x,y cos \alpha  z sin \alpha, y sin \alpha + z cos \alpha ) ## and so via small angle expansions have:
##y \to ( y(1\frac{\alpha^2}{2}+O(\alpha^4))z(\alpha\frac{\alpha^3}{3!}+O(\alpha^5))) ##
and
## z\to ( ( z(1\frac{\alpha^2}{2}+O(\alpha^4))+y(\alpha\frac{\alpha^3}{3!}+O(\alpha^5)))##
I am unsure now how expand. I thought perhaps a taylor expansion in multivariables  y and z was the idea, but I can't see how you would arrive at the answer attached with this:
Any tips appreciated.ta.
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