Editing Action (physics) (section)

=== Action of a generalized coordinate ===
A variable ''J<sub>k</sub>'' in the [[action-angle coordinates]], called the "action" of the generalized coordinate ''q<sub>k</sub>'', is defined by integrating a single generalized momentum around a closed path in [[phase space]], corresponding to rotating or oscillating motion:<ref name=Goldestein3>{{Cite book |last1=Goldstein |first1=Herbert |title=Classical mechanics |last2=Poole |first2=Charles P. |last3=Safko |first3=John L. |date=2008 |publisher=Addison Wesley |isbn=978-0-201-65702-9 |edition=3, [Nachdr.] |location=San Francisco Munich}}</ref>{{rp|454}}

<math display="block">
J_k = \oint p_k \,dq_k
</math>

The corresponding canonical variable conjugate to ''J<sub>k</sub>'' is its "angle" ''w<sub>k</sub>'', for reasons described more fully under [[action-angle coordinates]].  The integration is only over a single variable ''q<sub>k</sub>'' and, therefore, unlike the integrated [[dot product]] in the abbreviated action integral above.  The ''J<sub>k</sub>'' variable equals the change in ''S<sub>k</sub>''(''q<sub>k</sub>'') as ''q<sub>k</sub>'' is varied around the closed path.  For several physical systems of interest, J<sub>k</sub> is either a constant or varies very slowly; hence, the variable ''J<sub>k</sub>'' is often used in perturbation calculations and in determining [[adiabatic invariant]]s. For example, they are used in the calculation of planetary and satellite orbits.<ref name=Goldestein3/>{{rp|477}}