Betatron frequency and the Poincaré rotation number

Symplectic maps are routinely used to describe single-particle dynamics in circular accelerators. In the case of a linear accelerator map, the rotation number (the betatron frequency) can be easily calculated from the map itself. In the case of a symplectic nonlinear map, the rotation number is normally obtained numerically, by iterating the map for given initial conditions, or through a perturbation approach. Integrable maps, a subclass of symplectic maps, allow for an analytic evaluation of their rotation numbers. In this paper we propose an analytic expression to determine the rotation number for integrable symplectic maps of the plane, if an integral is explicitly known, and present several examples, relevant to accelerators. If the integral of motion is not explicitly known, one can obtain the rotation number numerically as outlined in Appendix B. These new results can be used to analyze the topology of the accelerator Hamiltonians as well as to serve as the starting point for a perturbation theory for maps.

Figure 1

Constant level sets of the integral $\mathcal{K}(q,p)=\text{const}$ (left). A particular curve representing a level set of $\mathcal{K}$ and several iterates of the map M (center). A three-dimensional phase space, $(q,p)+\mathrm{time}$, of the system (7) (right). Dark gray planes $t=0,\tau ,2\tau ,\dots$ represent the stroboscopic Poincaré section of the continuous flow of the system (red curve) which is identical to map M.

Figure 2

The partial action is defined as a sector area (blue) for one map iteration, divided by $2\pi$ (a). Convenient choices of the partial action for mappings in McMillan form: an area under the curve in II (blue) or IV (green) quadrants (b), and areas for initial conditions in a form of $(q_0,q_0)$ (c).

Figure 3

The top plot contains iterations (green dots) of the McMillan map ($a=1.6$, $b=1$). Constant level sets of the invariant are shown with blue lines. The bottom plot is the rotation number, Eq. (27), as a function of its integral, $\mathcal{K}$. The inset shows the linear approximation, Eq. (29).

Figure 4

Top row: Phase-space trajectories of a cubic map, obtained by tracking with $a=-0.85$ (left plot) and level sets of the approximate invariant (34) (right plot), on the same scale. The red and blue lines in the top left plot correspond to symmetry lines $p=q$ and $p=(aq+\epsilon q^3)/2$ respectively. Bottom row: The left plot shows the rotation number as a function of initial conditions in the form $q_0=p_0$, by using Eq. (2) (black solid line), and by using the Danilov equation, Eq. (12) numerically (orange dashed). The red solid line corresponds to the rotation number obtained from the approximate invariant (34) by using the Danilov equation as well. The right bottom plot shows the dependence of $\nu$ as a function of action $J$, from tracking (orange dashed) and from the approximate invariant (34) (red solid).

Figure 5

Radial (top) and angular (bottom) rotation numbers as a function of the first integral of the map, $\mathcal{K}$, for different values of its second integral, $p_{\theta }$ (shown with color labels). The map parameters are $b=1$ and $a=1.6$. Note that for $p_{\theta }=0$, ${\nu }_r=2{\nu }_{\theta }$, as expected, and equals the frequency $\nu$ from the one-dimensional example of Fig. 3.

Figure 6

Brown map. (a) Constant level sets of the invariant, $\mathcal{K}(q,p)=\text{const}$ (black solid polygons). Dashed black line $p=q$ and blue line $p=\frac{1}{2}|q|$ illustrate two reflection symmetries of the invariant polygons. Line segments are labeled with roman numerals. Green points are an example of a 9-cycle orbit, where the arabic numerals show the iteration number. (b) An example of possible contour of integration for the numerator and denominator in Danilov equation.