Editing Orbital elements (section)

=== Delaunay variables ===
The Delaunay orbital elements were introduced by [[Charles-Eugène Delaunay]] during his study of the motion of the [[Moon]].<ref name="Aubin-20142">{{cite book |last=Aubin |first=David |title=Biographical Encyclopedia of Astronomers |publisher=Springer New York |year=2014 |isbn=978-1-4419-9916-0 |place=New York City |pages=548–549 |chapter=Delaunay, Charles-Eugène |doi=10.1007/978-1-4419-9917-7_347}}</ref> Commonly called ''Delaunay variables'', they are a set of [[canonical variables]], which are [[action-angle coordinates]]. The angles are simple sums of some of the Keplerian angles, and are often referred to with different symbols than other in applications like so:

* the [[mean longitude]]: <math>\ell = L = M + \omega + \Omega</math>,
* the [[longitude of periapsis]]: <math>g = \varpi = \omega + \Omega</math>, and
* the [[longitude of the ascending node]]: <math>h = \Omega</math>

along with their respective [[Conjugate momentum|conjugate momenta]], ''{{mvar|L}}'', ''{{mvar|G}}'', and ''{{mvar|H}}''.<ref name="Shevchenko-20172">{{cite book |last=Shevchenko |first=Ivan |title=The Lidov–Kozai effect: applications in exoplanet research and dynamical astronomy |publisher=Springer |year=2017 |isbn=978-3-319-43522-0 |publication-place=Cham}}</ref> The momenta ''{{mvar|L}}'', ''{{mvar|G}}'', and ''{{mvar|H}}'' are the [[Action-angle coordinates|''action'' variables]] and are more elaborate combinations of the Keplerian elements ''{{mvar|a}}'', ''{{mvar|e}}'', and ''{{mvar|i}}''.

Delaunay variables are used to simplify perturbative calculations in celestial mechanics, for example while investigating the [[Kozai–Lidov oscillations]] in hierarchical triple systems.<ref name="Shevchenko-20172" /> The advantage of the Delaunay variables is that they remain well defined and non-singular (except for ''{{mvar|h}}'', which can be tolerated) even for circular and equatorial orbits.