Noncommutative Differential Calculus Approach to Discrete Symplectic Schemes on Regular Lattice

Guo, Hongxia; Wu, Ke; ShangHu, Wang; Wang, Shikun; Wei, Gong-Min

doi:10.1088/0253-6102/34/2/307

Commun. Theor. Phys.

2000

DOI: 10.1088/0253-6102/34/2/307

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Noncommutative Differential Calculus Approach to Discrete Symplectic Schemes on Regular Lattice

Hongxia Guo

Ke Wu

Wang ShangHu

et al.

Abstract: By means of a noncommutative differential calculus on function space of discrete Abelian groups and that of the regular lattice with equal spacing as well as the discrete symplectic geometry and a kind of classical mechanical systems with separable Hamiitonian of the type H(p, q) = T(p) + V(q) on regular lattice, we introduce the discrete symplectic algorithm, i.e., the phase-space discrete counterpart of the symplectic algorithm including original symplectic schemes and the jet-symplectic schemes in terms of … Show more

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Cited by 3 publications

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On variations in discrete mechanics and field theory

Guo

2003

Journal of Mathematical Physics

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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.

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On variations in discrete mechanics and field theory

Guo

2003

Journal of Mathematical Physics

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

Guo

et al. 2002

Commun. Theor. Phys.

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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.

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Symmetries and variational calculation of discrete Hamiltonian systems

Xia¹,

Chen²,

Fu³

et al. 2014

Chinese Phys. B

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We present a numerical simulation method of Noether and Lie symmetries for discrete Hamiltonian systems. The Noether and Lie symmetries for the systems are proposed by investigating the invariance properties of discrete Lagrangian in phase space. The numerical calculations of a two-degree-of-freedom nonlinear harmonic oscillator show that the difference discrete variational method preserves the exactness and the invariant quantity.

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Noncommutative Differential Calculus Approach to Discrete Symplectic Schemes on Regular Lattice

Cited by 3 publications

References 4 publications

On variations in discrete mechanics and field theory

On variations in discrete mechanics and field theory

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

Symmetries and variational calculation of discrete Hamiltonian systems

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