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:

Motion of a point under three 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?

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    $\begingroup$ It is not quite clear what you are asking for. If you just want to represent the result of a sequence of rotations in terms of a single rotation, this is an eigenvalue/eigenvector problem: any orientation-preserving linear transformation of a sphere to itself is equivalent to the rotation around certain axis by certain angle. But if you are actually interested in the visualization of the sequence of rotations as a motion on the sphere, no such a simplification exists: any motion on the sphere is a composition of (infinitesimal) rotations. $\endgroup$ Jun 15, 2020 at 9:37
  • $\begingroup$ I know that the question itself is somewhat unclear, but this is because I simply do not know how a simpler way of representing the evolution would look like -- hence the question. Maybe writing the motion as it is written above is in a way the simplest way of representing it. But I hoped that some decomposition into higher frequency rotations about three normalized axes or something like this could exist. $\endgroup$ Jun 17, 2020 at 6:51

1 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 $x$.

  • $\begingroup$ Thanks for your reply! I meant that infinitesimal rotations add in the sense that $R_{\boldsymbol{v}_1}(\mathrm{d}x)R_{\boldsymbol{v}_2}(\mathrm{d}x) = R_{\boldsymbol{v}_1 + \boldsymbol{v}_2}(\mathrm{d}x)$ when you see $\boldsymbol{v}$ as angular momentum. This is also evident from the BCH Formula because $T_{\boldsymbol{v}_1} + T_{\boldsymbol{v}_2} = T_{\boldsymbol{v}_1 + \boldsymbol{v}_2}$. I don't think that the BCH formula will help me in my endeavour, I hoped that there would me more something like "spherical Fourier series" or the like. $\endgroup$ Jun 11, 2020 at 6:59
  • $\begingroup$ @JohannesJakobMeyer: agreed, to first order they do add $\endgroup$
    – user21299
    Jun 12, 2020 at 1:24

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