关于旋进椭圆的讨论

根据Bertrand定理关于轨道闭合的条件,我们知道当中心引力的形式为径向距离的幂次型函数时,并不总能导致闭合轨道,即运动的质点不一定能返回到它自己的轨迹上去。这个定理指出:一切运动质点其初始条件稍微偏离圆轨道的要求条件时,只有当引力满足平方反比律(即牛顿万有引力定律)或线性定律(即虎克定律)时,轨道才是闭合的。 本文试图采用分析力学中作用角变量的方法,对一般情形下的轨道闭合条件进行讨论。这里我们看到使初始轨道接近于圆轨道的假设是不必要的,而且力的类型可推广到包括平方反比和其它幂次型的力。我们证明了轨道闭合的判别公式,并得到在上述情形下轨道非闭合的结论。 最后,如果中心引力包括有r~(-2)项和r~(-3)项时,象相对论性改正后所得到的那样,轨道就不再是闭合的,而是一旋进椭圆形轨道,椭圆的旋进角速度可以容易地计算出来。

THE DISCUSSION ABOUT PRECESSING ELLIPSE

According to the conditions for closed orbit which is usually called Bertrand's theorem, we know that the type of attractive central force varied as the power-law function of the radial distance does not always lead to closed orbits, i.e., the moving particle will not -always retrace its own footprint. The theorem points out that the only central forces which result in closed orbits for all moving particles whose initial conditions deviate more than slightly from the requirements for a circular orbits are the inverse square law, (i.e., Newton's law) and linear law, (i.e., Hooke's law).In this paper, we attempt to discuss the conditions of closed orbit in more general -cases on the basis of the method of action-angle variables in analytical mechanics. In this way, the supposition of the orbits near to a circular orbit is not necessary and the type of forces can be extended to the type of forces which include the forces of inverse square law and/or any power-law. We have given the critical formula of the closed orbits and have got conclusion of unclosed orbits in the above-mentioned conditions.Finally, if the central force contains an r-2 term and an r-3 term, such as is provided by relativistic corrections,then the orbit is no longer closed but is in the form of a processing ellipse. And the precessing angular velocity of the ellipse can be calculated easily.