A structure-preserving algorithm for time-scale non-shifted Hamiltonian 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 of Delta and del 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. (c) 2022 The Author(s). Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)