万有引力场中陀螺体的混沌运动

CHAOTIC MOTION OF GYROSTAT IN THE CENTRAL GRAVITATIONAL FIELD

研究万有引力场中沿圆轨道运行的非对称陀螺体的姿态运动．引入Ｄｅｐｒｉｔ正则变量建立系统的Ｈａｍｉｌｔｏｎ结构．利用Ｍｅｌｎｉｋｏｖ方法证明在万有引力矩作用下陀螺体产生混沌运动的可能性．对Ｐｏｉｎｃａｒｅ截面的数值计算表明提高陀螺体的转子转速可对混沌起抑制作用。

The attitude motion of an asymmetric gyrostat in a circular orbit subjected to the gravitational torque is investigated by using the version of Melnikov's method developed for a two degree of freedom Hamiltonian system with S^1 symmetry. For this purpose Deprit's variables are introduced to establish the Hamiltonian structure for this problem. The theoretical result on the chaotic motion of the gyrostat is verified by numerical computation of Poincare section. The study shows that: (a) By introducing Deprit's canonical variables, one can caress the Hamiltonian of the system of high order in a sample form, rendering it suitable for the application Of Melnikov's method. (b) The influence of the gravitational torque on the gyrostat can be regarded as perturbations to the torque-free motion of the gyrostat. When the angular momentum of the rotor h_2 is relatively small, there exist two saddle points in the phase plane for the torque-free motion of the gyrostat, which are connected by two heteroclinic orbits. Under the perturbation of the gravitational force, this highly degenerated structure breaks and chaos occurs near the heteroclinic orbits. As the perturbation increases, the chaotic area is enlarged, as shown in Figr.3(b), (c). It means that the motion of the system is chaotic in the sense of Smaie's horseshoe. (c) The rotor speed has obvious effect on the motion Of the gyrostat. As the rotor speed increases, the chaotic area gradually contracts and a chaotic motion will turn into a regular motion, as shown in Fign.3(c), (d). So the chaotic motion can be suppressed by increasing the rotor speed.