Variations in Discrete Mechanics and Field Theory

TL;DR

This paper explores variational principles in discrete mechanics and field theory, establishing conditions for symplectic preservation and conservation laws, and introduces a variable step-length approach based on Euler-Lagrange cohomology.

Contribution

It proposes a difference variational approach with variable step-length, extending Lee's method, and links Euler-Lagrange cohomology to symplectic and conservation properties in discrete systems.

Findings

Existence of nontrivial Euler-Lagrange cohomology in discrete and continuous systems.

Necessary and sufficient conditions for symplectic/multi-symplectic structure preservation.

Variable step-length determined by energy and momentum conservation relations.

Abstract

Some problems on variations are raised for classical discrete mechanics and field theory and the difference variational approach with variable step-length is proposed motivated by Lee's approach to discrete mechanics and the difference discrete variational principle for difference discrete mechanics and field theory on regular lattice. Based upon Hamilton's principle for the vertical variations and double operation of vertical exterior differential on action, it is shown that for both continuous and variable step-length difference cases there exists the nontrivial Euler-Lagrange cohomology as well as the necessary and sufficient condition for symplectic/multi-symplectic structure preserving properties is the relevant Euler-Lagrange 1-form is closed in both continuous and difference classical mechanics and field theory. While the horizontal variations give rise to the relevant identities or relations of the Euler-Lagrange equation and conservation law of the energy/energy-momentum tensor for continuous or discrete systems. The total variations are also discussed. Especially, for those discrete cases the variable step-length of the difference is determined by the relation between the Euler-Lagrange equation and conservation law of the energy/energy-momentum tensor. In addition, this approach together with difference version of the Euler-Lagrange cohomology can be applied not only to discrete Lagrangian formalism but also to the Hamiltonian formalism for difference mechanics and field theory.