Rotation| Rotation |
In General > s.a. kinematics
of special relativity; Reference Frame [rotating].
* In R3:
An element R of
SO(3), which can be parametrized by Euler angles, R(
,
, 
)
= R3() R1()
R3().
* Vector fields: In generalized
Cartesian coordinates,
1 =
(0; 0, z, –y) , 2
= (0; –z, 0, x) , 3
= (0; y, –x, 0) ;
in spherical coordinates,
1 =
(0; 0, sin, cot cos) , 2
= (0; 0, –cos, cot sin) , 3
= (0; 0, 0, –1) .
* Of a world-line: Its
meaning is well-defined, not just wrt 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 m 
m
ua = a (umAm),
m m va =
a (vmAm),
and m m wa =
a (wmAm),
where a is
the unit tangent to the world-line.
* 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.
Rotational Invariance in Physics > see hamiltonian dynamics; realism; spherical symmetry; [symmetry].
References > s.a. time [rotating clocks].
@ General references: Walker 90.
@ In general relativity:
Malament gq/00-in
[vs intuition]; Bel gq/03 [Wilson-Wilson,
Michelson-Morley experiments]; Kajari et al a0905-in.
@ Rotation of the universe: Hawking Obs(69);
Chaliasos ap/06 [rotation of galaxies and acceleration].
@ Teaching: Silva & Tavares AJP(07)jan
[angular momentum and angular velocity].
@ Related concepts
and systems: Lansey a0906/AJP
[rotations through imaginary angles]; > s.a. angular
momentum; stars.
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