# Composition of Lorentz pure rotations

1. Apr 2, 2009

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!

2. Apr 2, 2009

### 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)

3. Apr 2, 2009

### 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.

4. Apr 3, 2009

### George Jones

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