nLab Euler angle

Idea

The 3 classical Euler angles furnish a global parametrization and a smooth coordinate system on a big dense cell of the Lie group SO(3) of rotations in 3 dimensions, or on its double cover, $SU(2)$ group. Its essential feature is that it supplies a multiplicative decomposition of arbitrary rotation into 3 elements belonging to distinguished 1-parametric subgroups of rotations $SO(2) = U(1)$ around 2 coordinate axes, where the classical choices are z-x-z and the angular parameters for the elements in distinguished $SO(2)$ are the Euler angles.

Generalized constructions apply to some other Lie groups.

Introducing Euler angles in intrinsic interpretation

We consider rotations of a rigid body in $\mathbf{R}^3$ with origin $O$. For the definitions we do not need to specify the unit vectors, but only the orientation on the axes. We want to rotate a right handed triple of mutually orthogonal (semi)axes $OxOyOz$ into another, $Ox'Oy'Oz'$, by using rotations around coordinate axes. In this interpretation, the axes are around oriented axes which move together with the body. We first move around $Ox$, then around $Oz$ of the rotated body, and finally around $Ox$ of the rotated body.

The basic notion is the line of nodes $n$ which is the intersection of the plane $\langle x,y\rangle$ (generated by vectors along $x$ and along $y$) and plane $\langle x',y'\rangle$. Choose an orientation so that $OxOnOz$ be a right handed. Regarding that both $x$ and $n$ are orthogonal to $z$ there is an angle $\phi$, $|\phi|\leq\pi$, such that if we rotate around $Oz$ by angle $\phi$, $Ox$ becomes $On$, moreover this angle is unique except if $On = -Ox$, that is $|\phi| = \pi$. Now rotate around $Ox$ of the rotated body, which is the same as $On$ of the original coordinate system, so that $Oz$ becomes $Oz'$. This is possible because $On$ is orthogonal to both of them by construction. The angle of rotation is $\theta$, which is the same as the original angle between $Oz$ and $Oz'$. Finally, we rotate around $Oz$ of the new system (which is $Oz'$ of the original problem) by the angle $\psi$ between from $Ox$ (which is the line of nodes, $On$ of the fixed system) to $Ox'$, again observing the orientation. Regarding that $OxOz$ went into $Ox'Oz'$, the third semiaxis is automatically fixed.

Construction for $SU(2)$

(Following Vilenkin) Every $SU(2)$ matrix can be written in a form

$$u = \left(\array{ a & b \\ - \overline{b} & \overline{a} }\right)$$

where $a,b\in\mathbf{C}$, $det(u) = 1$ and $|a|^2 + |b|^2 = 1$. The simplest parametrization is to take $|i|, arg(a)$ and $arg(b)$ as independent parameters. If $a b \neq 0$ then there are unique parameters $\phi,\theta,\psi$ called Euler angles such that

$$|a| = cos\frac{\theta}{2} , \,\,\,\,\,\, arg(a) = \frac{\phi+\psi}{2}, \,\,\,\,\,\, arg(b) = \frac{\phi-\psi+\pi}{2}$$

and we can take $O\leq \phi\lt 2\pi$, $0\lt\theta\lt\pi$, $-2\pi\leq\psi\lt 2\pi$. If $a=0$ or $b = 0$ then we can find triples $(\phi,\theta,\psi)$ satisfying the above system but the choice is not necessarily unique. In any case, the matrix $u$ is now

$$u(\phi,\theta,\psi):= u = \left(\array{ cos\frac{\theta}{2} e^{i\frac{\phi+\psi}{2}}& i sin\frac\theta{2}e^{i\frac{\phi-\psi}{2}} \\ i sin\frac\theta{2}e^{i\frac{\psi-\phi}{2}} & cos\frac\theta{2} e^{-i\frac{\phi+\psi}{2}} }\right)$$

There is a decomposition

$$u(\phi,\theta,\psi) = \left(\array{ e^{\frac{i\phi}{2}}&0\\0&e^{-\frac{i\phi}{2}} }\right) \left(\array{ cos\frac\theta{2}&i sin\frac\theta{2}\\ i sin\frac\theta{2}&cos\frac\theta{2} }\right) \left(\array{ e^{\frac{i\psi}{2}}&0\\0&e^{-\frac{i\psi}{2}} }\right)$$

The standard homomorphism $SU(2)\to SO(3)$ sends this Euler angles to classical Euler angles for $SO(3)$; the map is surjective if we restrict to values $\theta\in[0,2\pi]$ as the classical Euler angle bounds suggest.

Related concepts

• Pauli matrix

References

• Wikipedia, Euler angles

• WolframMath, Euler angles

• Lev Landau, Mechanics

• Understanding Euler angles (robotics, engineerings and aviation conventions) web

• N. Ja. Vilenkin, Special functions and representation theory of groups $SO(n)$

• L. C. Biedenharn, J. D. Louck, The angular momentum in quantum physics, Enc. Math Appl. 8, Addison-Wesley 1981

• D. Brezov, C. Mladenova, I. Mladenov, Vector decompositions of rotations, J.Geom. Symmetry Phys. 28 (2012) 67-103; Vector parameters in classical hyperbolic geometry, Proc. 15th Int. Conf. on Geometry, Integrability and Quantization, Ivaïlo M. Mladenov, A. Ludu, A. Yoshioka, eds. (Sofia: Avangard Prima, 2014) 79-105 euclid

• S. Bertini, S. L. Cacciatori, B. L. Cerchiai, On the Euler angles for $SU(N)$, J. Math. Phys. 47:4, id.043510 (2006) doi arxiv:math-ph/0510075

• S. L. Cacciatori, B. L. Cerchiai, Euler angles for $G_2$, J. Math. Phys. 46, 083512 (2005) doi

• S. L. Cacciatori, F. Dalla Piazza, A. Scotti, Compact Lie groups: Euler constructions and generalized Dyson conjecture, Trans. Amer. Math. Soc. 369 (2017), 4709-4724 doi