Composition of Lorentz pure rotations

[amyadad]

Hello,

Given (in spherical coordinates) the resulting 4-vector K of the composition of 2 Lorentz pure rotations R1 and R2, where only R1 is known, I would like to find the angle of the "overall" rotation resulting from this composition.
In other words, I want to find the symbolic expression of the angle of the rotation R when :
R2.R1(U) = R(U) = K
and R2 is a known rotation and K is a known vector (R1, R unknown, U is a vector).

I first thought it would be easy but I have tried several equations and I have also tried to invert some of the rotations to simplify the equation but I should miss some point because I cannot find a reasonable expression (which I know do exist) for this angle.

Any help on that would be highly appreciated!

 

[xepma]

You actually don't need any reference to the Lorentz group - this is just the normal rotational group SO(3). There are quite some books that cover the parametrization of SO(3). I would suggest to start looking at the represenation of the group SO(3) in terms of so-called Euler angles. (just google it)

 

[Peeter]

In general the composition of two general boosts will not itself be a boost, but will be a composition of a boost and a spatial rotation.

The doc

http://faculty.luther.edu/~macdonal/GAGC/GAGC.html

under '2.4.4. Composition of boosts' contains a treatment of an algebraic split of such a boost composition into rapidity and rotation angles ... but perhaps somebody else has a reference for you that is free of the clifford algebra used there.

 

[George Jones]

Hello,

Given (in spherical coordinates) the resulting 4-vector K of the composition of 2 Lorentz pure rotations R1 and R2, where only R1 is known, I would like to find the angle of the "overall" rotation resulting from this composition.
In other words, I want to find the symbolic expression of the angle of the rotation R when :
R2.R1(U) = R(U) = K
and R2 is a known rotation and K is a known vector (R1, R unknown, U is a vector).

I first thought it would be easy but I have tried several equations and I have also tried to invert some of the rotations to simplify the equation but I should miss some point because I cannot find a reasonable expression (which I know do exist) for this angle.

Any help on that would be highly appreciated!

By "Lorentz pure rotations" do you mean spatial rotations, as xepma has interpreted, or boosts, as Peeter has interpreted?