Connected Bodies on the Orbit as Dynamic Billiard.
Abstract:
One considers the relative motion on the orbit with the blows of the two with a massless flexible unextensible tether connected point masses. These system models e.g. the dynamic of a spacecraft and a sound connected with a tether. The blow in this case means that the distance between the spacecraft and the sound is equal to the length of the tether and absolutely elastic reflection happens. One can treat such system as a dynamic billiard. The center of masses is assumed to be on a circular orbit. The motion of the sound relative to the center of masses can be presented as a free motion section series between the successive absolutely elastic blows. To obtain the Poincare section for the 'tied' motion one marks the trajectory lying on the definite energy level stroboscopically at the time of the blow. The corresponding sequence of the strobed points is shown in the Poincare section. The numerical investigations for various values of h reveal various stable n-unit periodic motions along with the regular (quasiperiodic) and chaotic ones show the evolution of the periodic motion 'islands' in the 'chaotic see' when h value changes as a parameter. It is essential that in the considered problem the energy value is a single parameter which determines the phase space structure.
