Editing Laplace–Runge–Lenz vector (section)

== Circular momentum hodographs ==
[[File:Kepler hodograph3.svg|thumb|right|280px|Figure 3: The momentum vector {{math|'''p'''}} (shown in blue) moves on a circle as the particle moves on an ellipse. The four labeled points correspond to those in Figure 1. The circle is centered on the {{mvar|y}}-axis at position {{math|''A''/''L''}} (shown in magenta), with radius {{math|''mk''/''L''}} (shown in green). The angle η determines the eccentricity {{mvar|e}} of the elliptical orbit ({{math|1=cos ''η'' = ''e''}}). By the [[inscribed angle|inscribed angle theorem]] for [[circle]]s, {{mvar|η}} is also the angle between any point on the circle and the two points of intersection with the {{math|''p''<sub>''x''</sub>}} axis, {{math|1=''p''<sub>''x''</sub> = ±''p''<sub>0</sub>}}, which only depend on {{mvar|E}}, but not {{mvar|L}}.]]

The conservation of the LRL vector {{math|'''A'''}} and angular momentum vector {{math|'''L'''}} is useful in showing that the momentum vector {{math|'''p'''}} moves on a [[circle]] under an inverse-square central force.<ref name="hamilton_1847_hodograph" /><ref name="goldstein_1975_1976" />

Taking the dot product of
<math display="block"> mk \hat{\mathbf{r}} = \mathbf{p} \times \mathbf{L} - \mathbf{A} </math>
with itself yields
<math display="block"> (mk)^2= A^2+ p^2 L^2 + 2 \mathbf{L} \cdot (\mathbf{p} \times \mathbf{A}). </math>

Further choosing {{math|'''L'''}} along the {{mvar|z}}-axis, and the major semiaxis as the {{mvar|x}}-axis, yields the locus equation for {{math|'''p'''}},
{{Equation box 1
|indent =:
|equation = <math> p_x^2 + \left(p_y - \frac A L \right)^2 = \left( \frac{mk} L \right)^2.</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F9FFF7}}

In other words, the momentum vector {{math|'''p'''}} is confined to a circle of radius {{math|1=''mk''/''L'' = ''L''/''ℓ''}} centered on {{math|(0, ''A''/''L'')}}.<ref>The conserved binormal Hamilton vector <math>\mathbf{B}\equiv \mathbf{L} \times \mathbf{A} / L^2</math> on this momentum plane (pink) has a simpler geometrical significance, and may actually supplant it, as <math>\mathbf{A} =\mathbf {B}\times \mathbf {L}</math>, see Patera, R. P. (1981). "Momentum-space derivation of the Runge-Lenz vector", ''Am. J. Phys'' '''49''' 593–594. It has length {{math|''A''/''L''}} and is discussed in section [[#Alternative scalings, symbols and formulations]].</ref> For bounded orbits, the eccentricity {{mvar|e}} corresponds to the cosine of the angle {{mvar|η}} shown in Figure 3.  For unbounded orbits, we have <math> A > m k</math> and so the circle does not intersect the <math>p_x</math>-axis.

In the degenerate limit of circular orbits, and thus vanishing {{math|'''A'''}}, the circle centers at the origin {{math|(0,0)}}.
For brevity, it is also useful to introduce the variable <math display="inline">p_0 = \sqrt{2m|E|}</math>.

This circular [[hodograph]] is useful in illustrating the symmetry of the Kepler problem.