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 $\nabla$ derivatives be beneficial to obtain higher convergence order in numerical analysis, but also it prompts the time-scale numerical computational scheme to have good properties, for instance, structure-preserving. In this paper, a structure-preserving algorithm for time-scale non-shifted Hamiltonian systems is proposed. By using the time-scale 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 of the time-scale non-shifted system as an example, it shows that the time-scale variational method is a structure-preserving algorithm with good stability and high computational accuracy. Unlike the traditional discrete differential variational principle with fixed step size, the time-scale discrete variational principle can also be calculated with variable and even segmented time step sizes