Symmetries And Structure-preserving Algorithms For Constrained Mechanical Systems On Time Scales

Not only can the theory of time scales calculus unify differential equations and difference equations,but also deal with quantum calculus when T＝q^N₀（q＞1） or T＝q^Z（?）｛0｝.This uniform approach means that complex new models with more variables can be established.In practical problems,the theory of time scales can also provide mathematical models for dynamic processes with continuous variables,discrete variables and piecewise continuous variables.In this paper,symmetries and structure-preserving algorithms for constrained mechanical systems on time scales are studied,including the fractional Noether symmetries for constrained mechanical systems on time scales,the structure-preserving algorithms for non-shifted systems on time scales,Herglotz-type Noether symmetries for non-conservative systems and Herglotz-type Noether symmetries for Birkhoffian systems on time scales.The details are as follows:1.Combining the fractional calculus with the time-scale calculus,fractional Noether symmetries for Lagrangian systems,Hamiltonian systems and Birkhoffian systems on time scales are investigated.According to Caputo（35）-type time-scale fractional variational principles,the time-scale fractional dynamics equations of the corresponding systems are derived.Furthermore,under the special infinitesimal transformations and general infinitesimal transformations,the definitions and criteria of Caputo（35）-type time-scale fractional Noether symmetries are given.The time-scale fractional Noether theorems under general infinitesimal transformations are obtained by the generalized Jost method.When α＝1,the fractional Noether theorem on time scales for a constrained mechanical system obtained becomes the integer-order Noether theorem on time scales in the corresponding system.When T＝R,the fractional Noether theorem on time scales for a constrained mechanical system is reduced to the continuous fractional Noether theorem in the corresponding system.The correctness of these theorems is verified by examples.2.The structural preserving algorithms for time-scale non-shifted Hamiltonian and Birkhoffian systems are studied.The difference between the time-scale non-shifted system and the time-scale shifted system is pointed out.Then,by the time-scale calculus of variations,the discrete time scale is taken from the variational principle of non-shifted systems,and the corresponding discrete variational principle is obtained.Furthermore,the differential dynamics equations and the symplectic numerical algorithms are derived.The definitions and criteria of the discrete Noether symmetries and quasi-symmetries are given by introducing the discrete infinitesimal transformations,and the discrete Noether theorems on time scales are obtained.The results of examples show that the time-scale discrete variational method is a structure-preserving algorithm.The new algorithm provides a numerical method for solving time-scale non-shifted dynamic equations and continuous equations.Unlike the traditional discrete variational method with fixed step sizes,the proposed method can also calculate with variable step sizes and even segmented time step sizes under certain conditions to improve the computational speed.3.The Herglotz-type variational principle and its dynamic equation of non-conservative Lagrange form are presented.The definition of Herglotz-type Noether symmetry is proposed and its Noether identity is given by introducing infinitesimal transformations.The Herglotz-type Killing equations are derived,and the Herglotz-type Noether theorem and its inverse theorem are established.The Herglotz-type Noether theorem becomes the classical Noether theorem for Lagrangian systems when the Herglotz-type Lagrangian does not contain z obviously.Furthermore,according to the Herglotz-type differential variational principle,the perturbation of Herglotz-type Noether symmetry under small perturbation is studied,and the adiabatic invariants of the system are obtained.The inverse perturbation problem of Herglotz-type Noether symmetry for non-conservative Lagrangian systems is also given.Combining with the time-scale calculus,the Herglotz-type variational principle for Birkhoffian systems on time scales is proposed,and the Herglotz-type Birkhoff’s equations on time scales are derived.Furthermore,the definition and criterion of Herglotz-type Noether symmetry for Birkhoffian systems on time scales are given.Then the conserved quantity of the system is found.The Noether identity and Herglotz-type Noether conserved quantity for Birkhoffian systems in continuous and discrete cases are given.In addition,under the corresponding transformations,the Herglotz-type Noether theorem for Birkhoffian systems on time scales can become the one for Hamiltonian systems or Lagrangian systems on time scales.