Variations in Discrete Mechanics and Field Theory

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.

29 pages

References in corpus (6)

• Nonperturbative Continuity in Graviton Mass versus Perturbative Discontinuity
• Total Variation in Hamiltonian Formalism and Symplectic-Energy integrators
• Difference Discrete Variational Principle,EULER-Lagrange Cohomology and Symplectic, Multisymplectic Structures
• Total Variation and Variational Symplectic-Energy-Momentum integrators
• Symplectic, Multisymplectic Structures and Euler-Lagrange Cohomology
• The Euler-Lagrange Cohomology Groups on Symplectic Manifolds

Cited by in corpus (4)

• Explicit high-order noncanonical symplectic algorithms for ideal two-fluid systems
• Difference Discrete Variational Principle in Discrete Mechanics and Symplectic Algorithm
• The Generalized Liouville's Theorems via Euler-Lagrange Cohomology Groups on Symplectic Manifold
• Discrete Formulation for the dynamics of rods deforming in space