Rotation |

**In General** > s.a. examples of lie groups [rotation groups SO(*n*)].

* __In R__

*

*ξ*_{1} =
(0; 0, *z*, –*y*) , *ξ*_{2}
= (0; –*z*, 0, *x*) , *ξ*_{3}
= (0; *y*, –*x*, 0) ;

in spherical coordinates,

*ξ*_{1} =
(0; 0, sin*φ*, cot*θ* cos*φ*) , *ξ*_{2}
= (0; 0, –cos*φ*, cot*θ* sin*φ*) , *ξ*_{3}
= (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 *u*^{a}, *v*^{a} and *w*^{a}
on the
world-line; The rate of change of these vectors measures the rotation;
There is no rotation iff *ξ*^{m} ∇_{m}
*u*^{a} = *ξ*^{a} (*u*_{m}*A*^{m}),
*ξ*^{m} ∇_{m} *v*^{a} =
*ξ*^{a} (*v*_{m}*A*^{m}),
and *ξ*^{m} ∇_{m} *w*^{a} =
*ξ*^{a} (*w*_{m}*A*^{m}),
where *ξ*^{a} is
the unit tangent to the world-line.

@ __General references__: Walker 90; O'Connell a1009-in [in
different physical theories].

> __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.

@ __Measurement__:
Wright et al PRL(13) [BEC-based rotation sensor]; Nolan et al PRA(16)-a1511 [spin-1 BEC in a ring trap].

@ __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.

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