The form invariance and the conserved quantity for a weakly nonholonomic system （WNS） are studied. The WNS is a nonholonomic system （NS） whose constraint equations contain a small parameter. The differential equations of motion of the system are established. The definition and the criterion of form invariance of the system are given. The conserved quantity deduced from the form invariance is obtained. Finally, an illustrative example is shown.
This paper studies the symmetry of Lagrangians of nonholonomic systems of non-Chetaev＇s type. First, the definition and the criterion of the symmetry of the system are given. Secondly, it obtains the condition under which there exists a conserved quantity and the form of the conserved quantity. Finally, an example is shown to illustrate the application of the result.
The characteristics of stationary and non-stationary skew-gradient systems are studied. The skew-gradient representations of holonomic systems, Birkhoffian systems, generalized Birkhoffian systems, and generalized Hamiltonian systems are given. The characteristics of skew-gradient systems are used to study integration and stability of the solution of constrained mechanical systems. Examples are given to illustrate applications of the result.
The problem of transforming autonomous systems into Birkhoffian systems is studied. A reasonable form of linear autonomous Birkhoff equations is given. By combining them with the undetermined tensor method, a necessary and sufficient condition for an autonomous system to have a representation in terms of linear autonomous Birkhoff equations is obtained. The methods of constructing Birkhoffian dynamical functions are given. Two examples are given to illustrate the application of the results.
The symmetry of Lagrangians of a holonomic variable mass system is studied. Firstly, the differential equations of motion of the system are established. Secondly, the definition and the criterion of the symmetry of the system are presented. Thirdly, the conditions under which there exists a conserved quantity deduced by the symmetry are obtained. The form of the conserved quantity is the same as that of the constant mass Lagrange system. Finally, an example is shown to illustrate the application of the result.
The method of nonholonomic mapping is utilized to construct a Riemann-Cartan space embedded into a known Riemann-Cartan space,which includes two special cases that a Weitzenbck space and a Riemann-Cartan space are respectively embedded into a Euclidean space and a Riemann space.By means of this mapping theory,the nonholonomic corresponding relation between the autoparallels of two Riemman-Cartan spaces is investigated.In particular,an autoparallel in a Riemann-Cartan space can be mapped into a geodesic line in a Riemann space and an autoparallel in Weitzenbck space be mapped into a geodesic line in Euclidean space.Based on the Lagrange-d'Alembert principle,the equations of motion for dynamical systems in Riemman-Cartan space should be autoparallel equations of the space.As applications,the problem of autoparallel motion of spinless particles,Chaplygin's nonholonomic systems and a rigid body rotating with a fixed point are investigated in space with torsion.
GUO YongXin1,LIU Chang2,WANG Yong3,LIU ShiXing1 & CHANG Peng2 1 College of Physics,Liaoning University,Shenyang 110036,China
