开尔文定理的一个注记及其应用

A NOTE OF THE THEOREM OF LORD KELVIN AND ITS APPLICATION

开尔文-台特-切塔耶夫定理中关于含保守力和陀螺力的线性系统稳定性条件指出:若保守系统不稳定,特征值无零根且具有偶数个正实根,则陀螺力的加入可使系统转为稳定.若正实根为奇数个,则加入陀螺力也不可能改变系统的不稳定性.定理不涉及有零根情形.作为开尔文定理的补充,本文基于二自由度系统证明:若保守系统不稳定,正实根为奇数个,且特征值存在零根,则陀螺力的加入也能使系统转为稳定.作为定理的实际应用,讨论匀速旋转的非惯性坐标系中的平衡稳定性问题.若将旋转坐标系中的离心惯性力视为有势力,则成为包含离心力势能的形式上的保守力场.由于受扰运动有科氏惯性力出现,此类系统与惯性参考系的保守力场并不等同,不允许应用拉格朗日定理以势能极小值条件判断其平衡稳定性.利用开尔文定理的上述补充条件,科氏惯性力的加入可改变此形式上保守系统的稳定性.以圆轨道二体系统和限制性三体问题的拉格朗日点的一次近似稳定性为例.仅考虑万有引力场和离心惯性力场,为含零根和一个正实根的不稳定保守系统.增加科氏惯性力项可使不稳定转为稳定.

According to the theorem of Lord Kelvin,an unstable conservative system can be stabilized by the gyroscopic force when the number of positive real characteristic roots is even without zero characteristic roots. In this paper,it is proved that an unstable conservative system with zero characteristic roots and an odd number of positive real characteristic roots can also be stabilized by the gyroscopic force. As an application of the theorem of Lord Kelvin,the stability problem in a non-inertial reference frame rotating with constant angular velocity is also discussed. The rotating coordinates system is regarded as a formal conservative system with potential energy including the field of centrifugal inertial force. But the rotating system is not equivalent to the conservative inertial system,because there exists the Coriolis inertial force in perturbed motion. Therefore,the Lagrange theorem of minimal potential energy is not allowed to determine the stability in the formal conservative system. However,when the complimentary condition of Kelvin＇s theorem is employed,the unstable formal conservative system can be stabilized by the Coriolis inertial force. The stability problems of two-body system with circle orbit and the Lagrange points of restrict three-body problem are analyzed as examples. Both systems under the action of gravitational and centrifugal force are unstable with one zero and one positive real characteristic root. When the Coriolis inertial force is considered,both unstable systems can be changed to be stable.