具有可积微分约束的力学系统的Lie对称性

LIE SYMMETRIES OF MECHANICAL SYSTEM WITH INTEGRAL DIFFERENTIAL CONSTRAINTS

研究具有可积微分约束的力学系统的Lie对称性与守恒星．采用两种方法：一是用不可积微分约束系统的方法；另一是用积分后降阶系统的方法．研究两种方法之间的关系．

It is well known that the Lie symmetry is an invariance of the ordinary differential equations under the infinitesimal transformations. The invariance of the equations of motion leads th satisfaction of the determining equations and the invariance of the equations of constraints leads the satisfaction of the restriction equations. A Lie symmetry can lead a conserved quantity under certain conditions. One of the conditions is the satisfaction of the structure equation. In this paper the Lie symmetries and conserved quantities of mechanical systems with integrable differential constraints are investigated. The integrable differential constraint is called semi-holonomic constraint.A mechanical system with integrable differential constraints can be considered as a nonholonomic system or as a holonomic system. We use two methods in studing the Lie symmetries and conserved quantities of the system. In the first method, the system is considered as a nonholonomic system and in the second method it is considered as a reduced holonomic system after integration.The definitions of weakly and strongly Lie symmetries in the two cases are given. The relation between two methods is obtained. The results prove that it is possible that some symmetries are lost in the second method.