[NOTES/CM-11003] Action Angle Variables

1. Action, angle variables

For a conservative system, the Hamiltonian is indeendent of time and Hamilton's characteristic function \(W\) obeys the Hamilton-Jacobi equation \begin{equation} H\Big(q,\pp[W]{q_k}\Big) =E. \end{equation} The solution depends on the coordinates and constants of integration \(\alpha_k, k = 1, ..., n\). Assuming that the Hamilton Jacobi equation is completely separable, we can write \begin{equation}\label{EQ02} W\equiv \sum_k W_k, \end{equation}
where each \(W_k\) is a function of a coordinate \(q_k\) and all constants \(\alpha_1 , \alpha_2 , ...\alpha_n\).

Since the momenta are given by \(p_k = \pp[W]{q_k}=\pp[W_k]{q_k}\), the functions \(W_k\) are given by the integrals of \(p_k\). \begin{equation}\label{EQ03} W_k = \int p_k (q_k , \alpha_1 , .., \alpha_n ) dq_k . \end{equation} A set of new variables called action variables \(J_k , k = 1, . . . , n\) are defined as \begin{equation}\label{EQ04} J_k =\frac{1}{2\pi}\oint p_k dq_k. \end{equation} where the integral runs over one cycle. Obviously the action variables are functions of the constants \(\alpha_1 , ..., \alpha_n\) and we write \begin{equation}\label{EQ05} J_k = J_k (\alpha_1 , ..., \alpha_n ), k = 1, 2..., n. \end{equation} Assuming that the above equations can be inverted and the constants \(\alpha_1 , ..., \alpha_n\) can be expressed in terms of the action variables \(J_k\). Thus the Jacobi complete integral, \eqref{EQ01}, can now be treated as a function of coordinates and the action variables. \begin{equation}\label{EQ06} W (q) = \sum_{k=1}^n W_k (q_k , J). \end{equation} and regard it as a generator of canonical transformation with the action variables \(J\) as the new coordinates. The angle variables, \(\phi_k\), are defined as variables conjugate to \(J_k\) and are given by \begin{equation} \phi_k = \pp[W]{J_k}. \end{equation} The process of separation of variables gives the Hamiltonian as a function of \(\alpha_k\) appearing as constants of separation. Thus we may now regard the Hamiltonian a function of action variables. \begin{equation}\label{EQ08} H = H(J_1 , ..., J_n ). \end{equation} The Hamilton’s equations for the system in action angle variables assume the form \begin{equation} \dd[J_k]{t} =-\pp[H]{W_k}=0, \qquad \dd[\phi_k]{t}= \pp[H]{J_k}\equiv \omega_k, \quad k=1,\ldots,n. \label{EQ10}\end{equation}  where \(ω_k = ω_k(J)\), being a function of \(J\)'s alone, are constants. The above equations have the solutions \begin{equation} J_k = \text{constant},\qquad \phi_k (t) = ω_k t + \phi_k (0). \end{equation}

2. Frequencies of bounded motion

Let us now consider a hypothetical cycle in which only one of the coordinates \(q_k\) changes with time and all other coordinates are held fixed. We shall now argue that the frequency \(ν_k\) of such a cycle is \begin{equation}\label{EQ11} \nu_k = \pp[H]{J_k} \end{equation} The change in \(W_k\) over this cycle is, by definition of \(J_k\), equal to \(2π J_k\). So using \eqref{EQ10} we have\begin{eqnarray} 2\pi J_k &=& \oint p_k dq_k =\oint J_k d\phi_k\label{EQ12}\\ &=& \oint J_k \dd[\phi]{t}\, dt=\oint J_k \pp[H]{J_k}\, dt\label{EQ13}\\ &=&J_k \tau_k \pp[H]{J_k}, \qquad \qquad(\because \text{integrand is a constant}) \label{EQ14} \end{eqnarray} where \(\tau_k\) is the time period of the cycle. Thus we get \begin{equation} \tau_k \, \pp[H]{J_k}= 2\pi \Longrightarrow \frac{2\pi}{\tau_k}= \pp[H]{J_k} \end{equation} The last equality says that the frequency \(ν_k\) associated with the hypothetical cyclic motion is \begin{equation} \boxed{\nu_k = \pp[H]{J_k}}. \end{equation}