Difference Discrete Variational Principles, Euler?Lagrange Cohomology and Symplectic, Multisymplectic Structures III: Application to Symplectic and Multisymplectic Algorithms

In the previous papers I and II, we have studied the difference discrete variational principle and the Euler?Lagrange cohomology in the framework of multi-parameter differential approach. We have gotten the difference discrete Euler?Lagrange equations and canonical ones for the difference discrete versions of classical mechanics and field theory as well as the difference discrete versions for the Euler?Lagrange cohomology and applied them to get the necessary and sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the Lagrangian and Hamiltonian formalisms. In this paper, we apply the difference discrete variational principle and Euler?Lagrange cohomological approach directly to the symplectic and multisymplectic algorithms. We will show that either Hamiltonian schemes or Lagrangian ones in both the symplectic and multisymplectic algorithms are variational integrators and their difference discrete symplectic structure-preserving properties can always be established not only in the solution space but also in the function space if and only if the related closed Euler?Lagrange cohomological conditions are satisfied.

Difference Discrete Variational Principle,Euler—Lagrange Cohomology and Symplectic,Multisymplectic Structures II：Euler—Lagrange Cohomology

In this second paper of a series of papers, we explore the difference discrete versions for the Euler?Lagrange cohomology and apply them to the symplectic or multisymplectic geometry and their preserving properties in both the Lagrangian and Hamiltonian formalisms for discrete mechanics and field theory in the framework of multi-parameter differential approach. In terms of the difference discrete Euler?Lagrange cohomological concepts, we show that the symplectic or multisymplectic geometry and their difference discrete structure-preserving properties can always be established not only in the solution spaces of the discrete Euler?Lagrange or canonical equations derived by the difference discrete variational principle but also in the function space in each case if and only if the relevant closed Euler?Lagrange cohomological conditions are satisfied.

A Note on Symplectic Algorithm

We present the symplectic algorithm in the Lagrangian formalism for the Hamiltonian systems by virtue of the noncommutative differential calculus with respect to the discrete time and the Euler-Lagrange cohomological concepts. We also show that the trapezoidal integrator is symplectic in certain sense.

On Symplectic and Multisymplectic Structures and Their Discrete Versions in Lagrangian Formalism

We introduce the Euler-Lagrange cohomology to study the symplectic and multisymplectic structures and their preserving properties in finite and infinite dimensional Lagrangian systems respectively. We also explore their certain difference discrete counterparts in the relevant regularly discretized finite and infinite dimensional Lagrangian systems by means of the difference discrete variational principle with the difference being regarded as an entire geometric object and the noncommutative differential calculus on regular lattice. In order to show that in all these cases the symplectic and multisymplectic preserving properties do not necessarily depend on the relevant Euler-Lagrange equations, the Euler-Lagrange cohomological concepts and content in the configuration space are employed.

变分和上同调的差分离散形式及其应用

回顾经典力学中的变分原理和离散变分原理的基本内容;简单介绍相空间作为辛流形上的欧拉-拉格朗日上同调、辛结构守恒的充要条件和刘维定理及其推广.在此基础上,介绍差分离散变分原理和欧拉-拉格朗日上同调的差分离散形式的概念和方法;及其在辛算法等领域的一些简单应用.

基于离散变分原理的Birkhoff辛算法

从Pfaff泛函变分的角度出发描述Birkhoff方程,再从变分离散的角度的到离散的Birkhoff方程,进而的到辛流形公式、构造辛算法,最后给出一个算例将其与传统的中点格式进行比较体现离散Birkhoff方程辛算法的优越性。