Study On Applications Of Lie Groups To Discrete Dynamical Systems

This dissertation is devoted to applications of Lie groups to discrete dymamicalsystems. We established and improved the symmetrical theories of discretedynamical systems, such as electromechanical dynamical systems, nonholonomicdynamical systems, discrete Hamiltonian systems, and discrete Birkhoff system. Thedissertation is divided into five parts:The first part consisted of Chapter1and Chapter2, surveys the applications of Liegroups, and presents the significance, main contents, and main innovations of thisdissertation. The Lie group transformation of discrete variables and the invariance oflattice equation are defined. The definition of the discrete Euler operator and thediscrete extremal equations are given.The second part of the dissertation includes Chapter3and Chapter4. Byintrouducing the infinitesimal transformations of time and generalized coordinates,the generalized Euler-Lagrange equation of discrete Chetaev nonholonomicdynamical systems with regular and irregular lattices is given. Based on theinvariance of the Hamilton action and the lattice equation under infinitesimaltransformation, the Noether identities, the Noether theorem and the Noetherconserved quantity are obtained associated with discrete Chetaev nonholonomicdynamical systems, then Noether theory of this system is constructed. Based on theinvariance of the equation of motion of system and the lattice equation under Lietransformation groups, the Noether identities, the determining equations of Liesymmetry, Lie symmetrical theorem and discrete Noether conserved quantity areobtained associated with discrete Chetaev nonholonomic dynamical systems, andthen the Lie symmetrical theory of discrete Chetaev nonholonomic dynamicalsystems is constructed. The form invariance of the Euler-Lagrange equation ofsystem under infinitesimal transformations is investigated. The definition, the criteria and the theorem of Mei symmetry are given. The Mei symmetrical theory of discreteChetaev nonholonomic dynamical systems is also constructed. By introuducing theinfinitesimal transformations with respect to time, generalized coordinates andgeneralized momentum, the Lie group theory is applied to discrete Hamiltoniansystems. Based on the invariance of the equation of motion of system under Lietransformation groups, the determining equations of Lie symmetry, Lie symmetricaltheorem and discrete conserved quantity are obtained，the Lie symmetrical theory ofdiscrete holonomic and nonholonomic non-conservative Hamiltonian systems areconstructed.The third part mainly contains Chapter5, deals with the applications of Lie groupsto electromechanical dynamical system, including the holonomic electromechanicaldynamical system and nonholonomic electromechanical system. The equations ofholonomic dynamical system are established in the regular lattice and irregularlattice. Based on the invariance of the Hamilton action and the lattice equation underinfinitesimal transformations, the Noether symmetrical theory of discrete holonomicelectromechanical system is constructed. The transformation operators ofdifferentiation corresponding to electromechanical system is introduced, based onthe invariance of the equation of motion of system and the lattice equation under Lietransformation groups, the determining equations of Lie symmetry, Lie symmetricaltheorem and discrete Noether conserved quantity are obtained associated withdiscrete holonomic electromechanical system. The Lie symmetrical theory ofdiscrete holonomic electromechanical system is constructed. Based on the equationof motion of nonholonomic electromechanical system (the Gaponov’s equation), thegeneralized Noether identities, the generalized Noether theorem and the generalizedNoether conserved quantity are obtained associated with the invariance of the extendHamilton action under infinitesimal transformations. The generalized Noether theoryof nonholonomic electromechanical systems is also constructed. Based on theinvariance of the equation of motion of nonholonomic electromechanical systems under Lie transformation groups, the determining equation and the structuralequation are obtained associated with nonholonomic electromechanical systems, andwe construct the Lie symmetrical theory and the inverse problems of nonholonomicelectromechanical systems. At the end of the chapter5, the equation of discretenonholonomic dynamical system is constructed, and the foundation theory ofNoether symmetry and the basic theory of Lie symmetry are presented for thediscrete nonholonomic dynamical system.The fourth part mainly contains Chapter6, deals with the applications of Liegroups to discrete Birkhoffian system. With the Pfaff action, we construct theNoether symmetrical theory of discrete Birkhoff system. Based on the invariance ofthe equation of motion of discrete Birkhoff system under Lie transformation groups,the determining equation and the structural equation are obtained associated withdiscrete Birkhoff system, and we construct the Lie symmetrical theory of discreteBirkhoff system. The relationship between Noether symmetry and Lie symmetry ofthe discrete Birkhoffian system is discussed.