The Study On Symmetries And Their Perturbation Theories For Constrained Mechanical Systems On Time Scales

The traditional methods of classical mechanics deal only with conservative systems,while almost all classical processes observed in the physical world are nonconservative.Hence,Researchers devoted themselves to the search for methods of dealing with friction and other forms of dissipation in classical and quantum mechanics.And in 1996,Riewe found that fractional order derivative was an effective method to deal with nonconservative forces.As a result,the study on dynamics of fractional constrained mechanical systems became a hot topic.A time scale is an arbitrary nonempty closed subset of the real numbers.It follows from this definition that time scale can unify continuous analysis,discrete analysis and quantum analysis,providing a powerful mathematical tool for the study of the complicated dynamical systems.Fractional order model and time scale calculus are used to study variational problem,symmetry and conserved quantity as well as perturbation to symmetry and adiabatic invariants in this paper.It follows from variational principle that Hamilton canonical equations with nabla derivatives on time scales,delta-nabla integral equations for Birkhoffian systems on time scales,generalized Birkhoff equations with delta derivatives on time scales,fractional generalized Birkhoff equations within Riemann-Liouville fractional derivatives,discrete fractional Lagrange equations and discrete fractional Birkhoff equations are established.Using symmetry methods,we obtain(1)Noether theorems for singular Lagrangian systems,Hamiltonian systems,Birkhoffian systems and generalized Birkhoffian systems on time scales;(2)Noether theorems for Birkhoffian systems within combined Riemann-Liouville fractional derivatives,Riesz-Riemann-Liouville fractional derivatives,combined Caputo fractional derivatives and Riesz-Caputo fractional derivatives;(3)Noether theorems for generalized Birkhoffian systems within Riemann-Liouville fractional derivatives;(4)Lie and Mei symmetry and conserved quantity for Birkhoffian systems on the basis of El-Nabulsi fractional model.Based on the definition of adiabatic invariants,we study(1)perturbation to Noether symmetry and adiabatic invariants for Lagrangian systems and nonholonomic systems on time scales;(2)perturbation to symmetry and adiabatic invariants for Birkhoffian systems in the sense of combined Riemann-Liouville fractional derivatives,Riesz-Riemann-Liouville fractional derivatives,combined Caputo fractional derivatives and Riesz-Caputo fractional derivatives;(3)perturbation to Noether quasi-symmetry and adiabatic invariants for generalized Birkhoffian systems within Riemann-Liouville fractional derivatives;(4)perturbation to symmetry and adiabatic invariants for fractional generalized Birkhoffian systems with variable order;(5)perturbation to Noether symmetry and adiabatic invariants for generalized Birkhoffian systems on the basis of El-Nabulsi fractional model;(6)perturbation to Lie and Mei symmetry and adiabatic invariants for Birkhoffian systems on the basis of El-Nabulsi fractional model.