Difference Discrete Variational Principle in Discrete Mechanics and Symplectic Algorithm**The project supported in part by National Natural Science Foundation of China under Grant Nos. 90103004 and 10171096 and the State Key Project for Basic Research of China under Grant No. G1998030601

Abstract

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 symplecticity 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.