Investigation Of Some Problems About Integration Theory Of Constraint Dynamical Systems

Focusing on the integral theory of constraint dynamical systems,three problems following are discussed in this paper:(1) the algebra structure and classical integral theory of the relative motion dynamical systems.(2) Lie symmetries and invariants as well as their converse problems of the constraint mechanical systems.(3) discrete variational principle,discrete Noether symmetries and discrete Lie symmetries of the constraint dynamical systems.The research development on integral theory of the constraint mechanical systems is introduced in chapter one(introduction),including non-Noether invariant theory,dynamics and integral theory of the relative motion,the history and recent state of the research about the symmetries and invariants of the discrete mechanical systems.Chapter two introduces Lie group of transformations and infinitesimal transformations, especially one parameter Lie groups of transformations,point transformations and extended transformations,which are the important mathematics foundations.Chapter three discusses following problems:(1) generalized force and generalized potential function of the inertial force.(2) Lagrange equations and Hamilton canonical equations of relative motion systems.(3) energy integral methods and mechanical energy conserved law.(4) relative motion Routh equations of 1-order nonlinear nonholonomic systems.(5)Discusses the algebra structure and integral methods of the dynamic equations of non conserved nonholonomic relative motion systems and indicates that the systems there are not only consistent algebra structure but also Lie admit algebra structure.Hence the classical Poisson integral methods can be partially used to find the first integrals of the non conserved nonholonomic relative motion systems.In chapter four,the Lie symmetries and its converse problems of Lagrange systems are discussed.Forward,the Lie symmetries and its converse problems of nonholonomic systems in terms of quasi-coordinates are discussed.The Lie symmetries and conserved quantities of constraint mechanical systems in phase space are first discussed and the Lie symmetries in canonical form are given.Then the Lie symmetry theory which is used to the dynamical systems is extended to the continuum systems,and the Lie symmetries of classic field are obtained.Finally, the Lie symmetries and conserved quantities of the constraint Hamilton systems are discussed. The restricted equations relating to the singular character are regarded as additional constraint equations.The dynamical canonical equations are established and their symmetries are discussed.Chapter five discusses the discrete symmetries of the constraint mechanical systems.First, the discrete variational principle and discrete dynamical equations are established.Then the discrete Noether symmetries and discrete first integrals are obtained.Finally,we take the lead in researching the discrete Lie symmetries and invariants of non conserved systems and want to put forward the research of the discrete symmetry theory.Chapter six summarizes the main results in this paper and talks about research plans in future.