13.3: Action-Angle Variables

For a closed one-dimensional system undergoing finite motion (essentially a bound state), the equations of motion can be reformulated using the action variable \(I=\frac{1}{2 \pi} \oint p d q\) in place of the energy \(E\). \(I\) is a function of energy alone in a closed one-dimensional system, and vice versa.

We’re visualizing here a particle moving back and forth in a one-dimensional well with potential zero at the origin, and the potential never decreasing on going out from the origin to infinity. Obviously, if a potential has two low points, local bound states can arise in different places, and the \(I, E\) relationship is complicated, with different branches, possibly coming together at high energies.

Caution

Notice the integral sign in the expression for the action variable \(I\) is \(\oint\) signifying an integral around a closed path, a circuit. Don’t confuse this integral with the abbreviated action integral, which has the same integrand, but is an integral \(\int_{0}^{q} p d q\) along a contour from a fixed starting point, say the origin, to the endpoint \(q\), not going around a closed path. (Apologies for using the same letter for the differential and the endpoint, just following Landau.)

In the spirit of the discussion of constants of motion above, we make a canonical transformation to \(I\) as the new “momentum”, using as generating function the abbreviated action \(S_{0}(q, I)\)

The original momentum

\begin{equation}
p=\left(\partial S_{0} / \partial q\right)_{E}=\left(\partial S_{0}(q, I) / \partial q\right)_{I}
\end{equation}

The new “coordinate” conjugate to the momentum \(I\) will be

\begin{equation}
w=\partial S_{0}(q, I) / \partial I
\end{equation}

This is called an angle variable, \(I\) is the action variable, they are canonical.

To find Hamilton’s equations in the transformed variables, since there is no time-dependence in the transformation, and the system is closed, the energy remains constant. Also, the energy is a function of \(I\) (meaning not of \(\omega\). )

Hence

\begin{equation}\dot{I}=\partial E(I) / \partial w=0, \quad \dot{w}=\partial E(I) / \partial I=d E(I) / d I\end{equation}

so the angle is a linear function of time: \(w=(d E / d I) t+\text { constant }\)

One further point about the action variable and the action: since we define the action as

\begin{equation}S_{0}(q, I)=\int_{0}^{q} p d q\end{equation}

it follows that if we track the change in this integral as time goes on and the system moves round and round the circuit in phase space, an additional term \(\Delta S_{0}=2 \pi I\) will be added to the action for each time round, so the action is multi-valued.