# Is there a "canonical" way to parameterize elements of $\operatorname{SO}(N)$?

In SO(3) the methods of parameterizing rotations in terms of sequential simple rotations are categorized as Euler angles or Tait—Bryan angles, with Euler angles often taken as the canonical way of writing rotations in physics. Is there a set of angles for N-dimensional rotations that has a similar canonical status in physics? In technical terms, I'm curious if there's a widely used method of parameterizing the $N$-dimensional, defining, representation of $\operatorname{SO}(N)$ in terms of rotation angles.

What I see mostly when dealing with $\operatorname{SO}(N)$ is a Lie-group generator centered parameterization like $$R = \exp\left(\omega_{ij}J^{ij}\right),$$ where $\omega_{ij}=-\omega_{ji}$ is the matrix of parameters, and the $J^{ij}$ are the $\frac{N\cdot(N-1)}{2}$ generator matrices; $J^{ij}$ for each $i$ and $j$ is a $d\times d$ matrix in the $d$-dimensional representation. This representation is convenient for many purposes, especially when dealing with different dimensional representations of the same group or when everything needed can be done with infinitesimal transformations, but it is not obvious what limits are needed on $\omega_{ij}$ to get a single covering of the set (except on sets zero measure that are generalizations of the poles that produce gimbal lock in 3-$d$).

Here is an example of an explicit paramterization of the defining representation of $\operatorname{SO}(N)$ that is a single cover of the group (includes domain restrictions on the angles), and has the volume element.

• Related questions: physics.stackexchange.com/q/371892 , physics.stackexchange.com/q/376901 Dec 29 '17 at 20:54
• Would Mathematics be a better home for this question? Related Math.SE questions: math.stackexchange.com/q/1364495 , math.stackexchange.com/q/1935384/11127 Dec 29 '17 at 21:01
• IIRC, you may take $\omega_{12}\in[0,2\pi)$ and the rest of angles in the interval $[0,\pi)$ to cover the whole manifold (but don't quote me on that). Note also, 10.1063/1.1666011 seems relevant. Dec 29 '17 at 21:01
• @Qmechanic Probably not, since I'm asking about if a widespread or most used parameterization exists in the physics world, not asking for a construction of one or the characterization of it's properties. Dec 29 '17 at 21:04
• A good reference which discusses explicitly the range of parameters is the book by: Murnaghan, F. D. (1962). The unitary and rotation groups (Vol. 3). Spartan Books If you are looking for parametrization and not so worried about the range of parameters then the discussion can be widen quite a bit. Dec 29 '17 at 21:45

[Here's a partial answer that doesn't deal with the range of the angles.]

There is no "canonical" parametrization but some are more convenient than others depending on your application.

A good reference is the book by: Murnaghan, F. D. (1962). The unitary and rotation groups (Vol. 3). Spartan Books. The key point in this book is that many parametrizations of the rotation groups can be obtained from a parameterization of the unitary group by removing phases. See also by the same author Murnaghan, Francis Dominic. On a convenient system of parameters for the unitary group. Proceedings of the National Academy of Sciences of the United States of America (1952): 127-129.

Given the observation above on the connection between parametrization of elements in the unitary and rotation group, there are a number of convenient choices.

The easiest ones in my opinion are by a sequence of adjacent rotations. In SO(4) and SO(5) this would be \begin{align} {\cal SO}(4)\sim &R_{12}(\theta_1)R_{23}(\theta_2)R_{34}(\theta_3)R_{12}(\theta_4)R_{23}(\theta_5)R_{12}(\theta_6)\\ {\cal SO}(5)\sim &R_{12}(\theta_1)R_{23}(\theta_2)R_{34}(\theta_3)R_{45}(\theta_4)\times {\cal SO}(4) \end{align} as a restriction of the parametrization of this arXiv submission. It has the advantage of using only adjacent transformations, i.e. $J^{i,i+1}$. Related to this are this scheme and the paper Reck, Michael, et al. Experimental realization of any discrete unitary operator. Physical Review Letters 73.1 (1994): 58 (which unfortunately is not apparently freely accessible on the web), although Reck et al use non-adjacent transformations.

There is also a nice parametrization in this paper which uses a different sequence of adjacent transformations, but more like $${\cal SO}(5)\sim R_{34}R_{45}R_{12}R_{23}R_{34}R_{45}R_{12}R_{23}R_{34}R_{12}$$ (the parameters are implicit in each $R_{ij}$). The unitary version is nice because (as discussed in the paper) it reduces the "optical depth" of the device and thus is very useful to minimize losses. I presume the rotation version would have the same property.

There is additional information available in the textbook by Robert Gilmore. This topic had its moments many years ago.

• I would categorize the "sequence of adjacent transforms" as Tait–Bryan-like. I would call a "mix the new dimension with every existing dimension (in order?)" strategy, like the parameterization I linked in the question, Euler-like. Dec 29 '17 at 23:23
• @SeanE.Lake see arxiv.org/abs/math-ph/0205016 (at least for unitaries) and also pdfs.semanticscholar.org/fabd/… (also for unitaries but could be restricted to rotations). Dec 30 '17 at 2:43

Just a small complement to the comments and answer to this important question.

Such a parametrization (with precise domain, see eq. (18) in reference below) was already given by Adolf Hurwitz in 1897 when he invented the Haar measure. The article is accessible through this link.

In the related unitary case, there is a detailed discussion of explicit Euler angles type parametrization in Section 2.3 of the recent book "Log-Gases and Random Matrices" by Peter Forrester.

• Very nice additional references, especially the Hurwitz paper... Dec 30 '17 at 18:26