Difference Discrete Variational Principles,Euler—Lagrange Cohomology and Symplectic,Multisymplectic Structures Ⅲ：Application to Symplectic and Multisymplectic Algorithms

In the previous papers I and II,we have studied the difference discrete variational principle and the Euler-Lagrange cohomology in the framework of multi-parameter differential approach.We have gotten the difference discrete Euler-Lagrange equations and canonical ones for the difference discrete versions of classical mechanics and field theory as well as the difference discrete versions for the Euler-Lagrange cohomology and applied them to get the necessary and sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the lagrangian and Hamiltonian formalisms.In this paper,we apply the difference discrete variational principle and Euler-Lagrange cohomological approach directly to the symplectic and multisymplectic algorithms.We will show that either Hamiltonian schemes of Lagrangian ones in both the symplectic and multisymplectic algorithms are variational integrators and their difference discrete symplectic structure-preserving properties can always be established not only in the solution space but also in the function space if and only if the related closed Euler-Lagrange cohomological conditions are satisfied.     

On Symplectic and Multisymplectic Structures and Their Discrete Versions in Lagrangian Formalism

We introduce the Euler-Lagrange cohomology to study the symplectic and multisymplectic structures and their preserving properties in finite and infinite dimensional Lagrangian systems respectively.We also explore their certain difference discrete counterparts in the relevant regularly discretized finite and infinite dimensional Lagrangian systems by means of the difference discrete variational principle with the difference being regarded as an entire grometric object and the noncommutative differential calculus on regular lattice.In order to show that in all these cases the symplectic and multisymplectic preserving properties do not necessarily depend on the relevant Euler-Lagrange equations,the Euler-Lagrange cohomological concepts and content in the configuration space are employed.     

Difference Discrete Variational Principles,Euler—Lagrange Cohomology and Symplectic,Multisymplectic Structures I：Difference Discrete Variational Principle

In this first paper of a series,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 noncommutative differential geometry.Regarding the difference as an entire geometric object,the difference discrete version of Legendre transformation can be introduced.By virtue of this variational principle,we can discretely deal with the variation problems in both the Lagrangian and Hamiltonican formalisms to get difference discrete Euler-Lagrange equations and canonical ones for the difference discrete versions of the classical mechanics and classical field theory.     

Noncommutative Differential Calculus and Its Application on Discrete Spaces

We present the noncommutative differential calculus on the function space of the infinite set and construct a homotopy operator to prove the analogue of the Poincare lemma for the difference complex. Then the horizontal and vertical complexes are introduced with the total differential map and vertical exterior derivative. As the application of the differential calculus, we derive the schemes with the conservation of symplecticity and energy for Hamiltonian system and a two-dimensional integral models with infinite sequence of conserved currents. Then an Euler-Lagrange cohomology with symplectic structure-preserving is given in the discrete classical mechanics.     

Difference Discrete Variational Principle in Discrete Mechanics and Symplectic Algorithm

We propose the difference discrete variational principle in discrete mechanics and symplectic algorithm with variable step-length of time in finite duration based upon a noncommutative differential calculus established in this paper. This approach keeps both symplecticity and energy conservation discretely. We show that there exists the discrete version of the Euler-Lagrange cohomology in these discrete systems. We also discuss the solution existence in finite time-length and its site density in continuous limit, and apply our approach to the pendulum with periodic perturbation. The numerical results are satisfactory.     

First integrals of the discrete nonconservative and nonholonomic systems

In this paper we show that the first integrals of the discrete equation of motion for nonconservative and nonholonomic mechanical systems can be determined explicitly by investigating the invariance properties of the discrete Lagrangian. The result obtained is a discrete analogue of the generalized theorem of Noether in the Calculus of variations.     

Optimized third-order force-gradient symplectic algorithms

With the natural splitting of a Hamiltonian system into kinetic energy and potential energy,we construct two new optimal thirdorder force-gradient symplectic algorithms in each of which the norm of fourth-order truncation errors is minimized.They are both not explicitly superior to their no-optimal counterparts in the numerical stability and the topology structure-preserving,but they are in the accuracy of energy on classical problems and in one of the energy eigenvalues for one-dimensional time-independent Schrdinger equations.In particular,they are much better than the optimal third-order non-gradient symplectic method.They also have an advantage over the fourth-order non-gradient symplectic integrator.     

Discrete gap breathers in a two-dimensional diatomic face-centered square lattice

In this paper we study the existence and stability of two-dimensional discrete gap breathers in a two-dimensional diatomic face-centered square lattice consisting of alternating light and heavy atoms, with on-site potential and coupling potential. This study is focused on two-dimensional breathers with their frequency in the gap that separates the acoustic and optical bands of the phonon spectrum. We demonstrate the possibility of the existence of two-dimensional gap breathers by using a numerical method. Six types of two-dimensional gap breathers are obtained, i.e., symmetric, mirror-symmetric and asymmetric, whether the center of the breather is on a light or a heavy atom. The difference between one-dimensional discrete gap breathers and two-dimensional discrete gap breathers is also discussed. We use Aubry's theory to analyze the stability of discrete gap breathers in the two-dimensional diatomic face-centered square lattice.     

Euler-Lagrange Forms and Cohomology Groups on Jet Bundles

Using the language of jet bundles, we generalize the definitions of Euler-Lagrange one-form and the associated cohomology which were introduced by Guo et al. [Commun. Theor. Phys. 37 (2002) 1]. Continuous and discrete Lagrange mechanics and field theory are presented. Higher order Euler-Lagrange cohomology groups are also introduced.     

Analysis of the tetraquark and hexaquark molecular states with the QCD sum rules

In this article, we construct the color-singlet-color-singlet type currents and the color-singlet-colorsinglet-color-singlet type currents to study the scalar D*■*, D*D* tetraquark molecular states and the vector D*D*■*, D*D*D* hexaquark molecular states with the QCD sum rules in details. In calculations, we choose the pertinent energy scales of the QCD spectral densities with the energy scale formula ■for the tetraquark and hexaquark molecular states respectively in a consistent way. We obtain stable QCD sum rules for the scalar D*■*, D*D*tetraquark molecular states and the vector D*D*■* hexaquark molecular state, but cannot obtain stable QCD sum rules for the vector D*D*D* hexaquark molecular state. The connected(nonfactorizable)Feynman diagrams at the tree level(or the lowest order) and their induced diagrams via substituting the quark lines make positive contributions for the scalar D*D* tetraquark molecular state, but make negative or destructive contributions for the vector D*D*D* hexaquark molecular state. It is of no use or meaningless to distinguish the factorizable and nonfactorizable properties of the Feynman diagrams in the color space in the operator product expansion so as to interpret them in terms of the hadronic observables, we can only obtain information about the short-distance and long-distance contributions.     

Difference Discrete Variational Principles, Euler-Lagrange Cohomology and Symplectic, Multisyrnplectic Structures Ⅲ: Application to Symplectic and Multisymplectic Algorithms

In the previous papers I and H, we have studied the difference discrete variational principle and the EulerLagrange cohomology in the framework of multi-parameter differential approach. W5 have gotten the difference discreteEulcr-Lagrangc equations and canonical ones for the difference discrete versions of classical mechanics and tield theoryas well as the difference discrete versions for the Euler-Lagrange cohomology and applied them to get the necessaryand sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the Lagrangianand Hamiltonian formalisms. In this paper, we apply the difference discrete variational principle and Euler-Lagrangecohomological approach directly to the symplectic and multisymplectic algorithms. We will show that either Hamiltonianschemes or Lagrangian ones in both the symplectic and multisymplectic algorithms arc variational integrators and theirdifference discrete symplectic structure-preserving properties can always be established not only in the solution spacebut also in the function space if and only if the related closed Euler Lagrange cohomological conditions are satisfied.     

欧拉-拉格朗日系统的jet辛算法

研究了在欧拉-拉格朗日系统上的jet辛算法.证明了第二作者在1998年给出的一个离散的欧拉-拉格朗日(DEL)方程存在一个离散形式的几何结构,它沿着解是不变的,这个结构可以通过对离散的作用量函数求导得到.由此,可以给出此格式的jet辛性质.利用这个结构证明了与此DEL方程相关的离散Nother定理.最后,给出了一个欧拉-拉格朗日方程上的jet辛差分格式的数值算例,并与其它的差分格式进行了比较.     

Non-uniqueness of solutions of Percival's Euler-Lagrange equation

Percival [5,6] introduced a Langrangian and an Euler-Lagrange equation for finding quasi-periodic orbits. In [3], we studied area preserving twist homeomorphisms of the annulus, using Percival's formalism. We showed that Percival's Lagrangian has a maximum on a suitable function space, and that a point where it takes its maximum is a solution of Percival's Euler-Lagrange equation. Moreover, in the rigorous interpretation of Percival's formalism which we gave in [3], the solutions of Percival's Euler-Lagrange equation correspond bijectively to a certain class of minimal sets. (We will prove this in Sect. 2.) In [4], we showed that Percival's Lagrangian takes its maximum at only one point. In this paper, we show that there existC area preserving twist diffeomorphisms of the annulus, for which there exists at least one solution of Percival's Euler-Lagrange equation where Percival's Lagrangian does not take its maximum. In other words, solutions of Percival's Euler-Lagrange equation need not be unique.Supported by NSF Grant No. MCS 79-02017     

Lagrangian submanifolds and the Euler-Lagrange equations in higher-order mechanics

A direct construction of the Euler-Lagrange equations in higher-order mechanics as a submanifold of a higher-order tangent bundle is given, starting from the Lagrangian submanifold defined by the Lagrangian function. This construction uses higher-order tangent bundle geometry, derives the Euler-Lagrange equations as the constraint equations of a submanifold, and makes no assumptions about the regularity of the Lagrangian.     

Multisymplectic Geometry,Local Conservation Laws and a Multisymplectic Integrator for the Zakharov–Kuznetsov Equation

The multisymplectic geometry for the Zakharov–Kuznetsov equation is presented in this Letter. The multisymplectic form and the local energy and momentum conservation laws are derived directly from the variational principle. Based on the multisymplectic Hamiltonian formulation, we derive a 36-point multisymplectic integrator.     

Splitting multisymplectic integrators for Maxwell’s equations

In the paper, we describe a novel kind of multisymplectic method for three-dimensional (3-D) Maxwell’s equations. Splitting the 3-D Maxwell’s equations into three local one-dimensional (LOD) equations, then applying a pair of symplectic Runge–Kutta methods to discretize each resulting LOD equation, it leads to splitting multisymplectic integrators. We say this kind of schemes to be LOD multisymplectic scheme (LOD-MS). The discrete conservation laws, convergence, dispersive relation, dissipation and stability are investigated for the schemes. Theoretical analysis shows that the schemes are unconditionally stable, non-dissipative, and of first order accuracy in time and second order accuracy in space. As a reduction, we also consider the application of LOD-MS to 2-D Maxwell’s equations. Numerical experiments match the theoretical results well. They illustrate that LOD-MS is not only efficient and simple in coding, but also has almost all the nature of multisymplectic integrators.     

The Hamiltonian Canonical Form for Euler-Lagrange Equations

Based on the theory of calculus of variation, some sufficient conditions are given for some Euler-Lagrange equations to be equivalently represented by finite or even infinite many Hamiltonian canonical equations. Meanwhile, some further applications for equations such as the KdV equation, MKdV equation, the general linear Euler-Lagrange equation and the cylindric shell equations are given.