非Четаев型非完整系统的Lie对称性与守恒量

LIE SYMMETRIES AND CONSERVED QUANTITIES OF NONHOLONOMIC SYSTEMS OF NON CHETAEV'S TYPE 1)

研究非Четаев型非完整系统的Ｌｉｅ对称性．首先利用微分方程在无限小变换下的不变性建立Ｌｉｅ对称所满足的确定方程和限制方程，给出结构方程并求出守恒量；其次研究上述问题的逆问题：根据已知积分求相应的Ｌｉｅ对称性；最后举例说明结果的应用．

There are two kinds of modern methods in finding conservation laws. One is the Noether method which is based on the invariance of the Hamilton's action under the infinitesimal transformations, the other one is the Lie method which is based on the invariance of the differential equations under infinitesimal transformations. In mathematics, the symmetry methods in the theory of differential equations have made great progress in recent years. In mechanics, the research on Lie symmetries and conservation laws of holonomic conservative systems just began in seventies. This paper involves Lie symmetries and conservation laws of nonholonomic sysems of non Chetaev's type. We studied two kinds of problems on Lie symmetries and conserved quantities. One is the direct problem of Lie symmetries:finding out the corresponding conserved quantity according to a given Lie symmetry. For this, the equations of motion of the systems must be established, then establishing the determining equations of Lie symmetries according to the Lie theory that making the differential equations invariant under infinitesimal transformations, then finding out Lie symmetries of the holonomic systems corresponding to the nonholonomic systems by solving the determining equations. In order to obtain Lie symmetries of the nonholonomic systems, it is necessary to make the equations of nonholonomic constraints invariant under infinitesimal transformations. And this is just the restriction equations. Lie symmetries may not bring about conservation laws. The proposition in this paper shows that, if the structure equation have solution with respect to Lie symmetries, then the conservation laws can be found. The another kinds of problem is called the inverse problem of Lie symmetries: find out the corresponding Lie symmetry according to a known conserved quantity. For this, Noether symmetry corresponding to the given conserved quantity need to be found by Noether theory, then examining the symmetry if it is or not Lie symmetry according to the determining equations and restriction equations. Since Chetaev's nonholonomic systems is a special case of the non Chetaev's type, the results in this paper have more universality.