The Research Of MNL Symmetry And New Conserved Quantity For Discrete Mechanical Systems

The theory of symmetries and conserved quantities of mechanical system occupies very important position in the mathematical disciplines. The research of conserved quantities has very important theoretical value and practical significance to understand the physical state and properties of the system. In order to solve problems of practical engineering by the theory of symmetries and conserved quantities, we need to discrete the continuous system. Based on the study of MNL symmetry(Mei symmetry, Noether symmetry, Lie symmetry) and conserved quantities in Lagrange system, we research the symmetries and a new conserved quantity for the discrete Lagrange system and Hamilton system by the difference variational method in this paper. Firstly, in configuration space and phase space using the differentical method obtain the differentical dynamical equations that retain the original structure of continuous equations of conserved mass system and variable mass system. Secondly, in infinitesimal group transformations the criterions of Mei symmetry, Noether symmetry and Lie symmetry for discrete mechanical system are given. The form and the condition of new conserved quantity directly deduced from Mei symmetry or indirectly from Noether symmetry and Lie symmetry. Finally, in this paper the researches are summarized, the researches on symmetry and new conserved quantities for discrete systems are prospected.