FIG. 1: The periodic solution of the system (2.2) obtained by numerical integration. The initial data are x$_{0}$ = (0 . 479000 , 0 . 000000) m, p$_{0}$ = (0 . 000000 , 1 . 290805) Jsm − 1 , the parameter κ 2 = 1 Jm − $^{2}$, and c = 1 ms − $^{1}$. (i.e. x = x $^{1}$), we get where the turning points are x = r$_{max}$ and x = − r$_{max}$ . On setting x$_{0}$ = − r$_{max}$ we can write the trajectory explicitly as where T = 4 r max c is the period of oscillations of a massless particle between the turning points x = r$_{max}$ and x = − r$_{max}$ , and [ a ] is the biggest integer in a . The trajectory (2.23) is 7 x ( t ) = ( − 1) [ 2 t T ] c { t − ( 2 [ 2 t T ] + 1 ) T 4 } , (2.23) x = ± ct + x$_{0}$, − r$_{max}$ ≤ x ≤ r$_{max}$, (2.22)