Rotational Periodicity

3. Each of the four coordinates, Eq.(), is the angular coordinate on one of the circles which makes up the torus, Eq.(34). Furthermore, the periodicity of each is measured in units of revolutions, one per period:

$$\left.\begin{array}{rl} \Delta\phi^u&=1\\ \Delta\phi^v&=1\\ \Delta\phi^x&=1\\ \Delta\phi^y&=1 \end{array}\right\}$$ (53)

The existence and the magnitude of four periods expressed by these equations is validated by the ensuing calculation. One takes advantage of the four-fold periodicity of the spacetime environment of the particle's world line. This periodicity is characterized by the four periods

$$\left.\begin{array}{rl} \Delta u&=\displaystyle \frac{2\pi}{\omeg... ...le \frac{2\pi}{\omega_2}\\ \Delta x&=L_x\\ \Delta y&=L_y \end{array}\right\} ~.$$ (54)

The transformation, Eq.(52), maps the given spacetime periods, Eq.(55), into the periods, Eq.(54), of the torus. For example, using Eqs.(52) and (42) one has

$$\Delta \phi^u = \frac{\partial W(u+2\pi/\omega_1,v,x,y;\mathbf{I})}{\partial I_u} -\frac{\partial W(u,v,x,y;\mathbf{I})}{\partial I_u}$$

$$=\frac{\partial}{\partial I_u}\left\{ \int_u^{u+2\pi/\omega_1} p_udu+0+0+0\right\}$$

$$=\frac{\partial I_u}{\partial I_u}=1$$

Analogous computations validate the remaining periodicities in Eq.(54).

For fixed action $\mathbf{I}=(I_u,I_v,I_x,I_y)$ the periodic angle coordinates $\{ \phi^u,\phi^v,\phi^x,\phi^y,\}$ span a four-dimensional torus, Eq.(34), while a given spacetime trajectory, Eqs.(48)-(51), is represented by a straight line winding on it. The relation between these toroidal coordinates and the physically given ones is

$$\phi^u = (u-u_0)\frac{\omega_1}{2\pi}$$

$$\phi^v = (v-v_0)\frac{\omega_2}{2\pi} +\frac{1}{4I_v^2}\frac{2\pi}{\omeg... ...L_x}I_x \cos \omega_1 u + \frac{\eta_y}{L_y}I_y \cos (\omega_1 u+\delta)\right.$$

$$~~~~~~~~~~~~~~~+\left.\frac{m}{8}\left( \eta_x^2 \sin 2\omega_1 u+ \eta_y^2 \sin (2\omega_1 u+\delta)\right) \right]^u_{u_0}$$

$$\phi^x = \frac{x-x_0}{L_x}-\frac{1}{4I_v}\frac{2\pi}{\omega_2}\frac{2m}{\omega_1} \left. \frac{\eta_x}{L_x}\cos \omega_1 u\right\vert^u_{u_0}$$

$$\phi^y = \frac{y-x_0}{L_y}-\frac{1}{4I_v}\frac{2\pi}{\omega_2}\frac{2m}{\omega_1} \left.\frac{\eta_y}{L_y}\cos (\omega_1 u+\delta) \right\vert^u_{u_0}~.$$

In the framework of this toroidal picture the physical coordinates are curvilinear, but the geometrical coordinates are rectilinear. Because of their geometrical simplicity, the action-angle variables are sometimes called normal coordinates.