Raw data were converted to three-dimensional rotation vectors, eye position vectors, and angular velocity vectors using methods that have been previously described [53].
The peculiar symmetry of the Kepler problem results in the conservation of both the angular momentum vector L and the LRL vector A (as defined above) and, quantum mechanically, ensures that the energy levels of hydrogen do not depend on the angular momentum quantum numbers l and m.
Rotation of the sphere and reprojection results in a continuous mapping of the elliptical orbits without changing the energy; quantum mechanically, this corresponds to a mixing of all orbitals of the same energy quantum number n. Valentine Bargmann noted subsequently that the Poisson brackets for the angular momentum vector L and the scaled LRL vector D formed the Lie algebra for SO(4).
We convert from our standard orbital parameters of i and (a,e,g,h) for the inner and outer orbit to orbital vectors i.e. eccentricity ê and angular momentum vector (hat{J_{mathrm{b}}}) for both orbits.
Without loss of the generality, three cross-orthogonal gyros in an IMU unit measure the angular rate vector ( {boldsymbol{upomega}}_{ib}^b ) while three accelerometers simultaneously measure the specific force vector ( {mathbf{f}}_{ib}^b ).
The plane of motion is perpendicular to the angular momentum vector L, which is constant; this may be expressed mathematically by the vector dot product equation r·L = 0; likewise, since A lies in that plane, A·L = 0.
Two sets of variables were introduced for Fukushima in 1994 by a canonical transformation and they are useful when the angle between z-satellite axis of a coordinate system fixed in artificial satellite and the rotational angular momentum vector is zero or when the angle between Z-equatorial axis and rotation angular momentum vector is zero.
The first half is devoted to a simultaneously sketchy and ponderous history of 20th-century physics, full of such technical jargon as "mixing angles," "perturbation expansions" and "conformal transformations," packed into sentences like "the operator corresponding to an angular momentum vector is precisely the one that implements an infinitesimal rotation about the axis given by that vector".
The quantization of the orientation of the angular momentum vector was confirmed in an experiment in 1922 by other German physicists, Otto Stern and Walther Gerlach.
