A structure-preserving algorithm for time-scale non-shiftedHamiltonian systems

The variational calculus of time-scale non-shifted systems includes both the traditional continuous and traditional significant discrete variational calculus.Not only can the combination ofand∇derivatives be beneficial to obtaining higher convergence order in numerical analysis,but also it prompts the timescale numerical computational scheme to have good properties,for instance,structure-preserving.In this letter,a structure-preserving algorithm for time-scale non-shifted Hamiltonian systems is proposed.By using the time-scale discrete variational method and calculus theory,and taking a discrete time scale in the variational principle of non-shifted Hamiltonian systems,the corresponding discrete Hamiltonian principle can be obtained.Furthermore,the time-scale discrete Hamilton difference equations,Noether theorem,and the symplectic scheme of discrete Hamiltonian systems are obtained.Finally,taking the Kepler problem and damped oscillator for time-scale non-shifted Hamiltonian systems as examples,they show that the time-scale discrete variational method is a structure-preserving algorithm.The new algorithm not only provides a numerical method for solving time-scale non-shifted dynamic equations but can be calculated with variable step sizes to improve the computational speed.

A Study on Time Scale Non-Shifted Hamiltonian Dynamics and Noether's Theorems

The time-scale non-shifted Hamiltonian dynamics are investigated,including both general holonomic systems and nonholonomic systems.The time-scale non-shifted Hamilton principle is presented and extended to nonconservative system,and the dynamic equations in Hamiltonian framework are deduced.The Noether symmetry,including its definition and criteria,for time-scale non-shifted Hamiltonian dynamics is put forward,and Noether's theorems for both holonomic and nonholonomic systems are presented and proved.The nonshifted Noether conservation laws are given.The validity of the theorems is verified by two examples.

Lie Symmetry Theorem for Nonshifted Birkhoffian Systems on Time Scales

The Lie theorem for Birkhoffian systems with timescale nonshifted variational problems are studied,including free Birkhoffian system,generalized Birkhoffian system and constrained Birkhoffian system.First,the time-scale nonshifted generalized Pfaff-Birkhoff principle is established,and the dynamical equations for three Birkhoffian systems under nonshifted variational problems are deduced.Afterwards,in the time-scale non-shifted variational problems,by introducing the infinitesimal transformations,Lie symmetry for free Birkhoffian system,generalized Birkhoffian system and constrained Birkhoffian system are defined respectively.Then Lie symmetry theorems for three kinds of Birkhoffian systems are deduced and proved.In the end,three examples are given to explain the applications for the results.