Simplifying the superposition of rotational motions

Question

I'm struggling to make sense of the following problem: Consider a sphere which undergoes a series of sequential rotations of the same angle $x$ about multiple axes identified by unit vectors $\{\boldsymbol{v}_j\}$. Then a vector $\boldsymbol{a}(0)$ representing a point on the sphere will move to $$ \boldsymbol{a}(x) = R_{\boldsymbol{v}_n}(x) \dots R_{\boldsymbol{v}_2}(x) R_{\boldsymbol{v}_1}(x)\boldsymbol{a}(0) $$

The motion the point is undergoing in terms of $x$ is rather chaotic, here an example I cooked up in Mathematica where a point is undergoing three distinct rotations:

While I understand that the rotation vectors will simply add up for an infinitesimal rotation $\mathrm{d}x$, it is unclear to me how to understand the superposition of those rotations in the general sense.

As this is a very common problem in the mechanics of rigid bodies, I'd expect that there exists some kind of machinery that allows for a better understanding of superposed rotations, but I have searched far and wide and didn't find anything suitable.

So my question is: Is there a way to simplify the superposition of multiple rotational motions of the above form? Can it be decomposed in a way (like Fourier coefficients in calculus maybe)? Is there a way to use constructions involving angular momentum and the like?

Answer 1

I don't think a simplification exists. (Also, infinitesimal rotations do not "add", except for rotations in 2 dimensions; see below.)

It may perhaps help to write your rotations in terms of the matrix exponential so $$R_{v}(x)=\exp(xT_v)$$ where $T_v$ is an antisymmetric 3x3 matrix corresponding to the element of the Lie algebra of the rotation group $SO(3)$ that gives rise to a rotation about the axis $v$.

Then the composition of rotations can in principle be calculated using the Baker-Campbell-Hausdorff formula (which will converge for small $x$). For two rotations already the formula gives $$ R_{v_1}(x)R_{v_2}(x)=\exp\Big(x(T_{v_1}+T_{v_2})+\frac{x^2}{2}[T_{v_1},T_{v_2}]+\frac{x^3}{12}([T_{v_1},[T_{v_1},T_{v_2}]]-[T_{v_2},[T_{v_1},T_{v_2}]]) +\dots\Big) $$ where the bracket is the matrix commutator. If $v_1$ and $v_2$ are not co-linear then the commutator $[T_{v_1},T_{v_2}]$ will not be, either. (In two dimensions, the same applies except the $T_v$ lies in the 1-dimensional space of antisymmetric 2x2 matrices. This corresponds to the fact there is only ever one axis of rotation in 2D.)

In principle the BCH formula gives the answer in terms of a rotation about a single axis and some angle, both of which are horrible nonlinear functions of both the individual axes and $x$.