标题： 差分变分原理，欧拉-拉格朗日上同调与辛，多辛结构 显示英文标题

标题： Difference Discrete Variational Principle,EULER-Lagrange Cohomology and Symplectic, Multisymplectic Structures

摘要： 我们研究了多参数微分方法框架下的差分离散变分原理，把前向差分视为非交换几何中的整个几何对象。借助这一变分原理，我们得到了经典力学和经典场论的差分离散版本的欧拉-拉格朗日方程以及哈密顿正则方程。我们还探讨了欧拉-拉格朗日上同调的差分离散版本，并将其应用于辛几何或多辛几何及其在拉格朗日形式和哈密顿形式下保持性质的研究。根据差分离散欧拉-拉格朗日上同调的概念，我们证明了辛几何或多辛几何及其差分离散结构保持性质不仅可以在由差分离散变分原理导出的离散欧拉-拉格朗日/正则方程的解空间中建立，而且在每种情况下都可以在函数空间中建立，当且仅当相关的闭欧拉-拉格朗日上同调条件成立时。我们还直接将差分离散变分原理和上同调方法应用于辛算法和多辛算法。 显示英文摘要

摘要： We study the difference discrete variational principle in the framework of multi-parameter differential approach by regarding the forward difference as an entire geometric object in view of noncomutative differential geometry. By virtue of this variational principle, we get the difference discrete Euler-Lagrange equations and canonical ones for the difference discrete versions of the classical mechanics and classical field theory. We also 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 Lagrangian and Hamiltonian formalism. 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/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. We also apply the difference discrete variational principle and cohomological approach directly to the symplectic and multisymplectic algorithms.