Research On Dynamical And Integral Theory For Nonholonomic Systems On Time Scales

The time scale is an arbitrary nonempty closed subset on the set of real numbers.The theory of mechanical systems on time scales unifies and expands the theory of continuous and discrete mechanical systems.It can not only reveal the distinguish and connections between the continuous and discrete dynamical systems,but also can more accurately describe the essence of the complex dynamical systems,and the two results of the difference equations and differential equations are avoided.Since the complexity of the time scales and the practical problems,the research on the theory for dynamical systems on time scales is still in the primary stage.Therefore,the problems of nonholonomic systems dynamics and their integral theories on time scales are also important aspect of the research on the fields of analytical mechanics.In this paper,based on the dynamics and integral theory for nonholonomic systems and the theory for mechanical systems on time scales,the variational principle for nonholonomic systems on time scales are established,the differential equations of motion for nonholonomic systems on time scales are derived,and the methods of reduction and the canonical transformation theory for mechanical systems on time scales are studied.The continuous and discrete dynamics and their integral theories for nonholonomic systems are two special cases of the research on the dynamical theory for nonholonomic systems on time scales.The main research work and achievements of the paper are as follows:1.The variational principles for the nonholonomic systems on time scales are researched.Firstly,the definitions and basic properties of calculus on time scales are briefly described.Secondly,the Euler-Lagrange form,Appell form and Nielsen form of d’Alembert-Lagrange principles on time scales are established.Finally,the exchange relations between the differential and variational operations of nonholonomic systems on time scales are deduced,and the variational principles for nonholonomic systems on time scales are established.2.The differential equations of motion for nonholonomic systems on time scales are established.Based on the d’Alembert-Lagrange principle and Lagrange multiplier method on time scales,the differential equations of motion with Multipliers and the corresponding generalized Chaplygin equations for nonholonomic systems on time scales are established.The Noether conservation quantity for generalized Chaplygin systems on time scales are obtained,the inner relationship between the Noether quasi-symmetries and the conserved quantities for generalized Chaplygin systems on time scales are established.3.The cycle integrals and methods of reduction for mechanical systems on time scales are proposed and studied.The cycle integrals for Lagrange systems,Hamilton systems and Chaplygin systems on time scales are given,and the differential equations of motion for the corresponding mechanical systems are reduced by using the cycle integrals on time scales.The results show that the form of the differential equations of motion for Lagrange systems,Hamilton systems and Chaplygin systems on time scales hold,but the numbers of the corresponding equations are reduced.4.The energy integral and methods of reduction for mechanical systems on time scales are proposed and studied.The generalized energy integrals for Lagrange systems,Hamilton systems and Chaplygin systems on time scales are given,and the differential equations of motion for the corresponding mechanical systems are reduced by using the generalized energy integrals on time scales.The results show that the form of the differential equations of motion for Lagrange systems,Hamilton systems and Chaplygin systems on time scales hold,but the numbers of the corresponding equations are reduced.5.The canonical transformations for a mechanical system on time scales are researched.We give the definition of the Poisson brackets,Jaccobi identities,and the Poisson brackets form of Hamilton canonical equations on time scales.And four basic forms of canonical transformations with nabla derivatives are established.And some examples are given to illustrate the application of the results and the role played by a generating function in the canonical transformations with nabla derivatives.