Action variable

With angle period $2\pi$, the action is the enclosed phase-space area divided by $2\pi$:

This is the action component of action-angle variables, rather than the entire coordinate construction.

For $H=p_r^2/2+{\ell }^2/(2r^2)-k/r$, a noncollision bound orbit has $\ell \ne 0$ and $-k^2/(2{\ell }^2)\le E<0$. Its radial action is $\sqrt{-2E}/\pi$ times ${\int }_{\alpha }^{\beta }\sqrt{(r-\alpha )(\beta -r)}dr/r$, where $\alpha ,\beta =[k\mp \sqrt{k^2+2E{\ell }^2}]/(-2E)$. Evaluating the integral gives the displayed action and $H=-k^2/[2(I_r+|\ell |{)}^2]$. The angular and radial frequencies have equal magnitudes, explaining closed orbits rather than generic quasiperiodicity. The circular limit has $I_r=0$; zero angular momentum gives collision in the unregularized phase space.

When a Hamiltonian parameter varies slowly compared with the orbital period and no separatrix is crossed, the action changes only at higher adiabatic order.

The action is an adiabatic invariant under these assumptions; other slowly varying systems can have different adiabatic invariants.

For a one-dimensional particle of energy $E$ between walls separated by $L$, the unnormalized action isAdiabatic conservation gives $E(t)L(t{)}^2=\text{constant}$.

If an action has parameter dependence $I\propto \sqrt{m/|E|}$, adiabatic conservation implies $|E|\propto m$.

When a gravitational potential changes slowly relative to an orbital period, the orbital actions remain constant provided no separatrix is crossed. In a spherical potential, angular momentum is conserved exactly and the radial action is adiabatically conserved.

Adiabatic invariance of an orbital action preserves action values during slow regular changes, but frequencies $\partial H/\partial J_i$ can vary. Closure requires rational frequency ratios, which need not persist. A slowly added harmonic component in a Kepler orbit potential produces generic apsidal precession.

For radial motion between turning radii $r_{-}$ and $r_{+}$ in a central potential, the radial action per unit mass isIt vanishes on a circular orbit and is canonically conjugate to the radial angle.

For a bound Kepler orbit of semimajor axis $a$ around mass $M$,Adiabatic central-mass change therefore preserves $Ma$ and orbital eccentricity.

For $H=p^2/(2m)-1/|q|=-|E|$, collision continuation between the two turning points gives $I=\sqrt{2m/|E|}$.