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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.
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.
(Following Vilenkin) Every $SU(2)$ matrix can be written in a form
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
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
There is a decomposition
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.
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 $SO(n)$
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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
Last revised on June 17, 2021 at 14:33:54. See the history of this page for a list of all contributions to it.