Editing Laplace–Runge–Lenz vector (section)

== Evolution under perturbed potentials ==
[[File:Relativistic precession.svg|right|thumb|250px|Figure 5: Gradually precessing elliptical orbit, with an eccentricity ''e''&nbsp;=&nbsp;0.667. Such precession arises in the Kepler problem if the attractive central force deviates slightly from an inverse-square law. The ''rate'' of precession can be calculated using the formulae in the text.]]

The Laplace–Runge–Lenz vector {{math|'''A'''}} is conserved only for a perfect inverse-square central force. In most practical problems such as planetary motion, however, the interaction potential energy between two bodies is not exactly an inverse square law, but may include an additional central force, a so-called ''perturbation'' described by a potential energy {{math|''h''(''r'')}}. In such cases, the LRL vector rotates slowly in the plane of the orbit, corresponding to a slow [[apsidal precession]] of the orbit.

By assumption, the perturbing potential {{math|''h''(''r'')}} is a conservative central force, which implies that the total energy {{mvar|E}} and angular momentum vector {{math|'''L'''}} are conserved. Thus, the motion still lies in a plane perpendicular to {{math|'''L'''}} and the magnitude {{mvar|A}} is conserved, from the equation {{math|1=''A''<sup>2</sup> = ''m''<sup>2</sup>''k''<sup>2</sup> + 2''mEL''<sup>2</sup>}}. The perturbation potential {{math|''h''(''r'')}} may be any sort of function, but should be significantly weaker than the main inverse-square force between the two bodies.

The ''rate'' at which the LRL vector rotates provides information about the perturbing potential {{math|''h''(''r'')}}. Using canonical perturbation theory and [[action-angle coordinates]], it is straightforward to show<ref name="goldstein_1980" /> that {{math|'''A'''}} rotates at a rate of,
<math display="block">\begin{align}
\frac{\partial}{\partial L} \langle h(r) \rangle & = \frac{\partial}{\partial L} \left\{ \frac{1}{T} \int_0^T h(r) \, dt \right\} \\[1em]
& = \frac{\partial}{\partial L} \left\{ \frac{m}{L^{2}} \int_0^{2\pi} r^2 h(r) \, d\theta \right\},
\end{align}</math>
where {{mvar|T}} is the orbital period, and the identity {{math|1=''L'' ''dt'' = ''m'' ''r''<sup>2</sup> ''dθ''}} was used to convert the time integral into an angular integral (Figure 5). The expression in angular brackets, {{math|{{langle}}''h''(''r''){{rangle}}}}, represents the perturbing potential, but ''averaged'' over one full period; that is, averaged over one full passage of the body around its orbit. Mathematically, this time average corresponds to the following quantity in curly braces. This averaging helps to suppress fluctuations in the rate of rotation.

This approach was used to help verify [[Albert Einstein|Einstein's]] theory of [[general relativity]], which adds a small effective inverse-cubic perturbation to the normal Newtonian gravitational potential,<ref name="einstein_1915">{{cite journal | last = Einstein | first = A. | author-link = Albert Einstein | date = 1915 | title = Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie | journal = Sitzungsberichte der Preussischen Akademie der Wissenschaften | volume = 1915 | pages = 831–839| bibcode = 1915SPAW.......831E }}</ref>
<math display="block">
h(r) = \frac{kL^{2}}{m^{2}c^{2}} \left( \frac{1}{r^{3}} \right).
</math>

Inserting this function into the integral and using the equation
<math display="block">
\frac{1}{r} = \frac{mk}{L^2} \left( 1 + \frac{A}{mk} \cos\theta \right)
</math>
to express {{mvar|r}} in terms of {{mvar|θ}}, the precession rate of the periapsis caused by this non-Newtonian perturbation is calculated to be<ref name="einstein_1915" />
<math display="block">
\frac{6 \pi k^2}{T L^2 c^2},
</math>
which closely matches the observed anomalous precession of [[Mercury (planet)|Mercury]]<ref>{{cite journal | last = Le Verrier | first = U. J. J. | author-link = Urbain Le Verrier | date = 1859 | title = Lettre de M. Le Verrier à M. Faye sur la Théorie de Mercure et sur le Mouvement du Périhélie de cette Planète | journal = Comptes Rendus de l'Académie des Sciences de Paris | volume = 49 | pages = 379–383}}</ref> and binary [[pulsar]]s.<ref>{{cite book | last = Will | first = C. M. | date = 1979 | title = General Relativity, an Einstein Century Survey | edition = SW Hawking and W Israel | publisher = Cambridge University Press | location = Cambridge | pages = Chapter 2 | no-pp = true}}</ref> This agreement with experiment is strong evidence for general relativity.<ref>{{cite book | last = Pais | first = A. | author-link = Abraham Pais | date = 1982 | title = Subtle is the Lord: The Science and the Life of Albert Einstein | url = https://archive.org/details/subtleislordscie00pais | url-access = registration | publisher = Oxford University Press }}</ref><ref>{{cite book | last = Roseveare | first = N. T. | date = 1982 | title = Mercury's Perihelion from Le Verrier to Einstein | url = https://archive.org/details/mercurysperiheli0000rose | url-access = registration | publisher = Oxford University Press| isbn = 978-0-19-858174-1 }}</ref>