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

TL;DR

This paper develops a difference discrete variational principle using noncommutative differential geometry, leading to discrete Euler-Lagrange equations and cohomology, which preserve symplectic and multisymplectic structures in discrete mechanics and field theories.

Contribution

It introduces a novel difference discrete variational framework with cohomological methods for preserving geometric structures in discrete systems.

Findings

Discrete Euler-Lagrange equations derived from the variational principle.

Symplectic and multisymplectic structures are preserved under certain cohomological conditions.

Application of the approach to symplectic and multisymplectic algorithms.

Abstract

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.