Mathematics

We consider the following problem: given two generators X, Y of the Lie algebra so(3) and a rotation g \in SO(3) we are looking for a factorization g=exp(t_1 A_1) ... exp(t_k A_k), with k \in N, t_i \in R, A_i \in {X,Y}, such that |t_1| + ... + |t_k| is minimal. If X,Y are perpendicular (i.e., the corresponding axes are perpendicular), then it is well known that every g \in SO(3) can be written as a product of at most three factors, the parameters t_1, t_2, t_3 being nothing but Euler angles. This raises the following questions: (1) Assume that X and Y are perpendicular and normalized. Do Euler angles provide optimal factorizations? (2) If not, which factorizations are optimal and how many factors are needed? (3) What is the situation like for SU(2)? (4) What happens if one considers generators which are not perpendicular to each other? Considering the factorization problem as a control problem on a Lie group we will obtain detailed answers to (1)--(3). In case of (4) we obtain results suggesting a quite natural conjecture.