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A quantity conserved to leading order when a parameter varies slowly compared with the natural motion period. The action variable of a regular periodic Hamiltonian orbit is a standard example, as explained by adiabatic invariance of the action.

The Hamiltonian is $\boxed{H(q,p)=p^2/(2m)+V(q)}$. On a regular family of closed phase-space trajectories, action-angle variables are canonical coordinates $(\theta ,I)$ in which $\theta$ is periodic modulo $2\pi$ and $H=H(I)$. Hamilton's equations then give $\dot{I}=0$ and $\dot{\theta }=dH/dI$.

The action variable isThe orientation is that of the motion, so the integral is the positive enclosed phase-space area divided by $2\pi$.

A constant-energy phase-space trajectory of the harmonic oscillator is the ellipseIts area is $\pi \sqrt{2E/(m{\omega }^2)}\sqrt{2mE}=2\pi E/\omega$. Thus the action and slow energy change areThe action variable is an adiabatic invariant when $|\dot{\omega }|\ll {\omega }^2$, with $\omega$ staying positive. Averaging over many short oscillations therefore keeps $I$ approximately fixed; an increase in frequency increases the energy in the same proportion.

Liouville integrability on a $2n$-dimensional symplectic manifold means that there are $n$ functionally independent first integrals, including $H$, whose pairwise Poisson brackets vanish. Near a compact connected regular common level set, the Arnold-Liouville theorem gives action-angle variables $(J_i,{\theta }_i)$ with $J_i=(2\pi {)}^{-1}\oint pdq$, ${\theta }_i$ periodic modulo $2\pi$, $H=H(J)$, ${\dot{J}}_i=0$ and ${\dot{\theta }}_i=\partial H/\partial J_i$.

For a frozen mass, the turning point is $q_{\ast }=(2E{)}^{1/(2k)}$. The action variable isAdiabatic invariance of the action holds to leading order when the mass varies slowly compared with an orbital period, away from degenerating orbits. Keeping $J$ fixed givesFor $k=1$ this agrees with $E=\omega J$ and $\omega \propto m^{-1/2}$.

LetAn orbit of energy $E>0$ is an ellipse in phase space, with intercepts $x=\pm \sqrt{2E/(m{\omega }^2)}$ and $p=\pm \sqrt{2mE}$. Its enclosed area is $2\pi E/\omega$, so the action variable isThe transformationis canonical and gives the action-angle variables. Substitution yields

Hamilton's equations give $\dot{q}=p$ and $\dot{p}=-{\omega }^{2}q$, hence $\ddot{q}+{\omega }^{2}q=0$. The phase-space energy curve is an ellipse. Its action variable isThus $\boxed{E=\omega I}$.

The symplectic area enclosed by a closed orbit in a two-dimensional phase plane. With the orientation of Hamiltonian motion it is $\oint pdq=2\pi I$, where $I$ is the action variable.

An integral curve of a vector field in phase space, representing the positions and momenta of a moving system. A periodic one-dimensional Hamiltonian orbit encloses the area used to define the action variable.