Difference Discrete Variational Principle in Discrete Mechanics and   Symplectic Algorithm
  We propose the difference discrete variational principle in discrete mechanics and symplectic algorithm with variable step-length of time in finite duration based upon a noncommutative differential calculus established in this paper. This approach keeps both symplicticity and energy conservation discretely. We show that there exists the discrete version of the Euler-Lagrange cohomology in these discrete systems. We also discuss the solution existence in finite time-length and its site density in continuous limit, and apply our approach to the pendulum with periodic perturbation. The numerical results are satisfactory. 
Xu-Dong Luo, Han-Ying Guo, Yu-Qi Li, Ke Wu