Symmetry And Conservation Of Discrete Hamilton System Research

It has been a long time to use Lie group theory to study the differential equations,such as solving differential equations, classifying differential equations andestablishing properties of solution space of the differential equations and so on. Inrecent years, the theory has been also used to research discrete equations, mainly thedifference equations and differential-difference equations. In this paper, we use theinfinitesimal Lie group transformation to study the symmetries and first integrals ofdiscrete nonconservative and nonholonomic Hamilton systems.For the discrete nonconservative Hamilton systems, we defined the Hamiltonaction, suggested the variational principle for the systems and demonstrated therelation between the total variation and the discrete virtual placement. At last, thediscrete Hamiltonian canonical equations and discrete energy equations can bederived. According to the quasi-invariance of discrete Hamiltonian action andequation of lattice under the infinitesimal transformation, Noether’s identity ofdiscrete nonconservative Hamilton systems, Noether’s theorem and discrete Noether’sconservation can be obtained. We perfectly established the Noether symmetry theoryfor the discrete nonconservative Hamilton systems.In the discrete nonconservative Hamilton systems, we also introduced the Lietransformation group, and constructed the discrete difference operator correspondingto the Lie transformation group. The determining equations can be obtained withrespect to the invariance of the Hamiltonian canonical equations and discrete energyequations under the infinitesimal transformation. Then, we gave the Lie symmetrytheorem, and deduced the discrete Noether’s conservation, and perfectly establishedthe Lie symmetry theory for the discrete nonconservative Hamilton systems.For the discrete nonholonomic Hamilton systems, the variational principle ofHamilton action and the relation between the total variation and the discrete virtualplacement are given, and the discrete Hamiltonian canonical equations and discreteenergy equations can be derived. According to the quasi-invariance of discrete Hamiltonian action and equation of lattice under the infinitesimal transformation, wederived the discrete Noether’s identity, and researched the Noether theorem of thediscrete nonholonomic Hamilton system corresponding to the holonomic one. We asleresearched the generalized Noether theorem and the corresponding discrete Noether’sconservation, and perfectly established the Noether symmetry theory for the discretenonholonomic Hamilton systems.In the discrete nonholonomic Hamilton systems, we defined the Lietransformation operator. The determining equations can be obtained with respect tothe invariance of the Hamiltonian canonical equations and discrete energy equationsunder the infinitesimal transformation. We researched the Lie symmetry theorem ofthe discrete nonholonomic Hamiltonian systems corresponding to the holonomicHamiltonian systems. Ulteriorly, we researched the weak Lie symmetrytransformations and the strong Lie symmetry transformations, and perfectlyestablished the Lie symmetry theory for the discrete nonholonomic Hamilton systems.The novelty of this thesis consists in deriving the discrete Hamilton canonicaland energy equation of the discrete nonconservative and nonholonomic Hamiltonsystems, and establishing the Noether symmetry theory and Lie symmetry theory forthe discrete nonconservative and nonholonomic Hamilton systems.