Study On Symmeties And Conserved Quantities Of Dynamical Systems Based On The Variational Integrators

The symmetries and conserved quantities of dynamical systems based on thediscrete variational integrators are studied while the discrete variational principle areconsidered. The numerical computation method based on the variational integratorsis a symplectic method. And the theory of symmetries and conserved quantitiesbased on the discrete variational integrators can provide the right solutions of theequations of the dynamical systems. Therefore, the theory of symmetries andconserved quantities of dynamical systems based on the discrete variationalintegrators are important theoretical and practical significance. The main content ofthis paper can be summarized as follows:The first part consists of Chapter1and Chapter2: it summarizes the the currentstate of the research of the discrete symmetries and conserved quantities ofdynamical systems. And it presents the significance and the current stateof numerical computation method based on the variational integrators, main contents,and main innovations of this dissertation. Two main types of discretization are given.The difference equtions of the Hamiltonian system are derived corresponding to thedifference discrete variational principle and the discrete variational principle,respectly.The second part consists of Chapter3：the three types of discrete equations andthe symplectic schemes of the Hamilton systems are presented according to threetypes of discrete Legendre transformation. The physical interpretation of the threekinds of discrete Hamilton equations and the symplectic scheme are analized. Thedifferent forms of Noether theorem and conserved quantity are constructed based onthree types of discrete Legendre transformation.The third part consists of Chapter4：the discrete energy equations and themoment equations of the discrete systems in field are presented. The discreteanalogue of Noether-type identities in field theory is investigated by means ofdifference discrete variational principle with the difference being regarded as an entire geometric object. The discrete analogue of Noether theorems is obtained. It isproved that the example of the discretization for the nonlinear Schr dinger equationillustrates that there exists the discrete analogue of Noether conservation laws whenthe symmtries are given in field theory.The fourth part deals with the discrete non-Noether symmetries of thenonconsertive Hamiltonian systems. This part consists of Chapter5, Chapter6andChapter7. By introuducing the infinitesimal transformations of the time andgeneralized coordinates, the generalized Euler-Lagrange equation, the Lie symmetryand Mei symmetry of the discrete equation of the nonconsertive Hamiltoniansystems are derived while the second type of discretization and the Legendretransformations are considered. The definitions and the criterions of the Liesymmetry and Mei symmetry are derived. The Noether conserved quantities areobtained according to the discrete Noether theorem. The numerical calculations ofthe Kepler system show the difference discrete variational method preserves theexactness and the invariant quantity. This example also shows the discrete resultsagree to the continuous ones. The conformal invariance of Mei symmetry andconserved quantities for discrete Lagrangian systems under the infinitesimaltransformation of Lie groups are studied. The definition of conformal invarianceof Mei symmetry for discrete Lagrangian systems is constructed, together with itscriterion equations and determining equations. The conserved quantities of thesystems are presented using the structure equation satisfied by the gaugefunction.In the last part, the research of the whole work is summarized. The innovationpoints are refined and the future work is planned.