Birkhoff系统的一类积分不变量的构造

分别建立了自由Birkhoff系统和约束Birkhoff系统的非等时变分方程，并且利用系统的Birkhoff方程及其非等时变分方程证明，可由第一积分直接构造该系统基于非等时变分的一类积分不变量。文中，举例说明结果的应用。

THE CONSTRVCTION OF A CLASS OF INTEGRAL INVARIANTS FOR BIRKHOFFIAN SYSTEM

There is an intimate connection between the first integral and the integral invariant of a mechanical system, and Whittaker indicated that by using a first integral one can construct an integral invariant for holonomic conservative system. This paper studies the first integral and the construction of integral invariants for Birkhoffian systems. Firstly, a free Birkhoffian system is discussed. By using the relation of simultaneous variation and nonsimultaneous variation, the paper established the equations of nonsimultaneous variation for the free Birkhoffian system. Making use of the equations of Birkhoffian of the systems and their equations of nonsimultaneous variation, the paper have proved that an integral invariant based on the nonsimultaneous variation can be directly constructed by a first integral of the systems, and the inverse proposition is also held. Secondly, a constrained Birkhoffian system is considered. The equations of nonsimultaneous variation for the constrained Birkhoffian system is established. According to the equations of nonsimultaneous variation and the differential equations of motion of the constrained Birkhoffian system, the paper have proved the same results as the free Birkhoffian system. Two examples are given to illustrate the application of the results.