Noether Symmetry And Conserved Quantity Of Nonholonomic Controllable Mechanical Systems In Phase Space On Time Scales

The motion of a mechanical system is related to the forces acting on it and the constraints imposed on it.Therefore,motion can be controlled by forces or by constrainst.With the development of modern engineering mechanics,control theory has been widely used.It is of great significance to study non holonomic controllable mechanical systems.In this paper,based on the theory of time scale,Noether's theorems for nonholonomic controllable mechanical systems and second-order linear nonholonomic controllable mechanical systems in phase space are established,respectively.Further,Noether theory for a weakly nonholonomic system on time scales are investigated.1.By introducing the generalized momentum and Hamilton function on time scales,and based on the Hamilton principle with nonpotential in phase space the equations of motion for the system is established.We offered the definitions and criteria of the generalized Noether quasi-symmetry,and the expression of the corresponding conserved quantity is given.2.On the basis of time scales,the Hamilton equation for second-order linear nonholonomic controllable mechanical systems on time scale is established.Consider a one-parameter family of infinitesimal transformations with respect to the time,and based on the generalized quasi invariance.We offered the definitions and criteria of the generalize Noether quasi-symmetry,the corresponding conserved quantity is obtained.3.Based on the principle of Hamilton,the dynamic equations of motion for the system are given.we provide the differential equations of motion for the first-order approximate system corresponding to the weakly nonholonomic system.We offered the definition and criteria of Noether symmetry for the first-order approximate system of the weakly nonholonomic system.Then,the expression of the approximate conserved quantity of Noether symmetry in weakly nonholonomic systems is obtained.