Difference Discrete Variational Principle in Discrete Mechanics and Symplectic Algorithm

We propose the difference discrete variational principle indiscrete mechanics and symplectic algorithm with variablestep-length of time in finite duration based upon a noncommutativedifferential calculus established in this paper. This approachkeeps both symplecticity and energy conservation discretely. We show that there exists the discrete version of theEuler-Lagrange cohomology in these discrete systems. We alsodiscuss the solution existence in finite time-length and its sitedensity in continuous limit, and apply our approach to the pendulum with periodic perturbation.The numerical results are satisfactory.