Rotation

In General > s.a. examples of lie groups [rotation groups SO(n)].
* In R3: An element R of SO(3), which can be parametrized by Euler angles, R(φ, θ, ψ) = R3(ψ) R1(θ) R3(φ).
* Vector fields: In generalized Cartesian coordinates, a set of generators of SO(3) is

ξ1 = (0; 0, z, −y) ,       ξ2 = (0; −z, 0, x) ,       ξ3 = (0; y, −x, 0) ;

in spherical coordinates,

ξ1 = (0; 0, sinφ, cotθ cosφ) ,       ξ2 = (0; 0, −cosφ, cotθ sinφ) ,       ξ3 = (0; 0, 0, −1) .

* Of a world-line: Its meaning is well-defined, not just with respect to something – it can be measured by a spiked sphere with springs and beads; To define it quantitatively, introduce 3 orthogonal vectors ua, va and wa on the world-line; The rate of change of these vectors measures the rotation; There is no rotation iff ξmm ua = ξa (umAm), ξmm va = ξa (vmAm), and ξmm wa = ξa (wmAm), where ξa is the unit tangent to the world-line.
@ General references: Walker 90; O'Connell in(10)-a1009 [in different physical theories].
@ Teaching: Wheatland et al AJP(21)mar [demos with mobile phones, principal axes].
> Related topics: see mach's principle [rotation problem]; Newton's Bucket [rotations and absolute space]; Reference Frame [rotating].

As a Dynamical Process
* Stationary rotations: The rotation of a free generic three-dimensional rigid body is stationary if and only if it is a rotation around one of three principal axes of inertia, assumed to be distinct (if a moment of inertia is degenerate, rotation is stationary around any rotation axis in the corresponding eigensubspace).
* Measurement: The most sensitive instruments are laser gyroscopes, and atom interferometers; The latter have sensitivities of one-hundredth of a degree/min [@ Lenef et al PRL(97) + pn(97)feb], and potentially much less; > s.a. Gyroscope.
@ Measurement: Wright et al PRL(13) [BEC-based rotation sensor]; Nolan et al PRA(16)-a1511 [spin-1 BEC in a ring trap]; > s.a. Detectors.
@ In general relativity: Malament gq/00-fs [vs intuition]; Bel gq/03 [Wilson-Wilson, Michelson-Morley experiments]; Kajari et al proc(09)-a0905; Klioner et al IAU(09)-a1001 [relativistic aspects of the rotation of celestial objects].
@ In astrophysics and cosmology: Hawking Obs(69) [of the universe]; Chaliasos ap/06 [rotation of galaxies and acceleration]; Iorio JCAP(10)-a1004 [rotation of distant masses, solar-system constraints]; > s.a. galaxies [including rotation curves]; star properties.
@ Teaching: Silva & Tavares AJP(07)jan [angular momentum and angular velocity].
@ Examples: news PhysOrg(10)sep [fastest-spinning macroscopic object, graphene flake at 60 Mrpm].
@ Variations: Lansey a0906/AJP [rotations through imaginary angles]; Izosimov JPA(12)-a1202 [stationary rotations in higher dimensions].
> Related topics: see angular momentum; kinematics of special relativity; Moment of Inertia; time [rotating clocks].
> Rotational invariance: see hamiltonian dynamics; realism; spherical symmetry / symmetry.
> In quantum theory: see quantum oscillators.