Editing Laplace–Runge–Lenz vector (section)

== Rotational symmetry in four dimensions ==
[[File:Kepler Fock projection.svg|thumb|right|300px|Figure 8: The momentum hodographs of Figure 7 correspond to stereographic projections of [[great circle]]s on the three-dimensional {{mvar|η}} unit sphere. All of the great circles intersect the {{math|''η<sub>x</sub>''}} axis, which is perpendicular to the page; the projection is from the North pole (the {{math|'''w'''}} unit vector) to the {{math|''η''<sub>''x''</sub>}}−{{math|''η''<sub>''y''</sub>}} plane, as shown here for the magenta hodograph by the dashed black lines. The great circle at a latitude {{mvar|α}} corresponds to an [[eccentricity (mathematics)|eccentricity]] {{math|1=''e'' = sin ''α''}}. The colors of the great circles shown here correspond to their matching hodographs in Figure 7.]]

The connection between the Kepler problem and four-dimensional rotational symmetry {{math|SO(4)}} can be readily visualized.<ref name="bander_itzykson_1966" /><ref name="rogers_1973">{{cite journal | last = Rogers | first = H. H. | date = 1973 | title = Symmetry transformations of the classical Kepler problem | journal = Journal of Mathematical Physics | volume = 14 | issue = 8 | pages = 1125–1129 | doi = 10.1063/1.1666448|bibcode = 1973JMP....14.1125R }}</ref><ref>{{cite book | last = Guillemin | first = V. |author2=Sternberg S. | date = 1990 | title = Variations on a Theme by Kepler | publisher = American Mathematical Society Colloquium Publications | volume = 42 | isbn = 0-8218-1042-1}}</ref> Let the four-dimensional Cartesian coordinates be denoted {{math|(''w'', ''x'', ''y'', ''z'')}} where {{math|(''x'', ''y'', ''z'')}} represent the Cartesian coordinates of the normal position vector {{math|'''r'''}}. The three-dimensional momentum vector {{math|'''p'''}} is associated with a four-dimensional vector <math>\boldsymbol\eta</math> on a three-dimensional unit sphere
<math display="block">\begin{align}
\boldsymbol\eta & = \frac{p^2 - p_0^2}{p^2 + p_0^2} \mathbf{\hat{w}} + \frac{2 p_0}{p^2 + p_0^2} \mathbf{p} \\[1em]
& = \frac{mk - r p_0^2}{mk} \mathbf{\hat{w}} + \frac{rp_0}{mk} \mathbf{p},
\end{align}</math>
where <math>\mathbf{\hat{w}}</math> is the unit vector along the new {{mvar|w}} axis. The transformation mapping {{math|'''p'''}} to {{math|'''η'''}} can be uniquely inverted; for example, the {{mvar|x}} component of the momentum equals
<math display="block"> p_x = p_0 \frac{\eta_x}{1 - \eta_w}, </math>
and similarly for {{math|''p<sub>y</sub>''}} and {{math|''p<sub>z</sub>''}}. In other words, the three-dimensional vector {{math|'''p'''}} is a stereographic projection of the four-dimensional <math>\boldsymbol\eta</math> vector, scaled by {{math|''p''<sub>0</sub>}} (Figure 8).

Without loss of generality, we may eliminate the normal rotational symmetry by choosing the Cartesian coordinates such that the {{mvar|z}} axis is aligned with the angular momentum vector {{math|'''L'''}} and the momentum hodographs are aligned as they are in Figure 7, with the centers of the circles on the {{mvar|y}} axis. Since the motion is planar, and {{math|'''p'''}} and {{math|'''L'''}} are perpendicular, {{math|1=''p<sub>z</sub>'' = ''η''<sub>''z''</sub> = 0}} and attention may be restricted to the three-dimensional vector {{nowrap|1=<math>\boldsymbol\eta = (\eta_w, \eta_x, \eta_y)</math>.}} The family of [[Apollonian circles]] of momentum hodographs (Figure 7) correspond to a family of [[great circle]]s on the three-dimensional <math>\boldsymbol\eta</math> sphere, all of which intersect the {{math|''η''<sub>''x''</sub>}} axis at the two foci {{math|1=''η''<sub>''x''</sub> = ±1}}, corresponding to the momentum hodograph foci at {{math|1=''p<sub>x</sub>'' = ±''p''<sub>0</sub>}}. These great circles are related by a simple rotation about the {{math|''η''<sub>''x''</sub>}}-axis (Figure 8). This rotational symmetry transforms all the orbits of the same energy into one another; however, such a rotation is orthogonal to the usual three-dimensional rotations, since it transforms the fourth dimension {{math|''η''<sub>''w''</sub>}}. This higher symmetry is characteristic of the Kepler problem and corresponds to the conservation of the LRL vector.

An elegant [[action-angle variables]] solution for the Kepler problem can be obtained by eliminating the redundant four-dimensional coordinates <math>\boldsymbol\eta</math> in favor of elliptic cylindrical coordinates {{math|(''χ'', ''ψ'', ''φ'')}}<ref>{{cite journal | last = Lakshmanan | first = M. |author2=Hasegawa H. | title = On the canonical equivalence of the Kepler problem in coordinate and momentum spaces | journal = Journal of Physics A | volume = 17 | issue = 16 | pages = L889–L893 | doi = 10.1088/0305-4470/17/16/006 | date = 1984|bibcode = 1984JPhA...17L.889L }}</ref>
<math display="block">\begin{align}
\eta_w &= \operatorname{cn} \chi \operatorname{cn} \psi, \\[1ex]
\eta_x &= \operatorname{sn} \chi \operatorname{dn} \psi \cos \phi, \\[1ex]
\eta_y &= \operatorname{sn} \chi \operatorname{dn} \psi \sin \phi, \\[1ex]
\eta_z &= \operatorname{dn} \chi \operatorname{sn} \psi,
\end{align}</math>
where {{math|sn}}, {{math|cn}} and {{math|dn}} are [[Jacobi's elliptic functions]].