nLab
Euler angle

Contents

Contents

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)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)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)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 R 3\mathbf{R}^3 with origin OO. 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 OxOyOzOxOyOz into another, OxOyOzOx'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 OxOx, then around OzOz of the rotated body, and finally around OxOx of the rotated body.

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

Construction for SU(2)SU(2)

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

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

where a,bCa,b\in\mathbf{C}, det(u)=1det(u) = 1 and |a| 2+|b| 2=1|a|^2 + |b|^2 = 1. The simplest parametrization is to take |i|,arg(a)|i|, arg(a) and arg(b)arg(b) as independent parameters. If ab0a b \neq 0 then there are unique parameters ϕ,θ,ψ\phi,\theta,\psi called Euler angles such that

|a|=cosθ2,arg(a)=ϕ+ψ2,arg(b)=ϕψ+π2 |a| = cos\frac{\theta}{2} , \,\,\,\,\,\, arg(a) = \frac{\phi+\psi}{2}, \,\,\,\,\,\, arg(b) = \frac{\phi-\psi+\pi}{2}

and we can take Oϕ<2πO\leq \phi\lt 2\pi, 0<θ<π0\lt\theta\lt\pi, 2πψ<2π-2\pi\leq\psi\lt 2\pi. If a=0a=0 or b=0b = 0 then we can find triples (ϕ,θ,ψ)(\phi,\theta,\psi) satisfying the above system but the choice is not necessarily unique. In any case, the matrix uu is now

u(ϕ,θ,ψ):=u=(cosθ2e iϕ+ψ2 isinθ2e iϕψ2 isinθ2e iψϕ2 cosθ2e iϕ+ψ2) 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(ϕ,θ,ψ)=(e iϕ2 0 0 e iϕ2)(cosθ2 isinθ2 isinθ2 cosθ2)(e iψ2 0 0 e iψ2) 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)SO(3)SU(2)\to SO(3) sends this Euler angles to classical Euler angles for SO(3)SO(3); the map is surjective if we restrict to values θ[0,2π]\theta\in[0,2\pi] as the classical Euler angle bounds suggest.

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)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)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 2G_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 9, 2020 at 19:33:44. See the history of this page for a list of all contributions to it.