Difference Discrete Variational Principle in Discrete Mechanics and Symplectic Algorithm

TL;DR

This paper introduces a difference discrete variational principle for discrete mechanics and symplectic algorithms with variable time steps, ensuring symplecticity and energy conservation, and demonstrates its effectiveness through numerical experiments.

Contribution

It develops a novel difference discrete variational principle based on noncommutative calculus, preserving symplecticity and energy in variable step-length discrete systems.

Findings

Maintains symplecticity and energy conservation discretely.

Establishes discrete Euler-Lagrange cohomology.

Numerical results confirm effectiveness on pendulum with perturbation.

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 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.