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Difference Discrete Variational Principle,Euler—Lagrange Cohomology and Symplectic,Multisymplectic Structures II:Euler—Lagrange Cohomology 被引量:9
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作者 GUOHan-Ying WUKe 等 《Communications in Theoretical Physics》 SCIE CAS CSCD 2002年第2期129-138,共10页
In this second paper of a series of papers, we explore the difference discrete versions for the Euler?Lagrange cohomology and apply them to the symplectic or multisymplectic geometry and their preserving properties in... In this second paper of a series of papers, we 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 the Lagrangian and Hamiltonian formalisms for discrete mechanics and field theory in the framework of multi-parameter differential approach. 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 or 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. 展开更多
关键词 discrete variation Euler-Lagrange cohomology symplectic and multisymplectic structures
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On Symplectic and Multisymplectic Structures and Their Discrete Versions in Lagrangian Formalism 被引量:4
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作者 GUOHan-Ying LIYu-Qi 《Communications in Theoretical Physics》 SCIE CAS CSCD 2001年第6期703-710,共8页
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 the... 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 geometric 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. 展开更多
关键词 Euler-Lagrange cohomology difference discrete variational principle symplectic structure
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Symplectic Schemes for Birkhoffian System 被引量:8
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作者 SUHong-Ling QINMeng-Zhao 《Communications in Theoretical Physics》 SCIE CAS CSCD 2004年第3期329-334,共6页
A universal symplectic structure for a Newtonian system including nonconservative cases can be constructed in the framework of Birkhoffian generalization of Hamiltonian mechanics. In this paper the symplectic geometry... A universal symplectic structure for a Newtonian system including nonconservative cases can be constructed in the framework of Birkhoffian generalization of Hamiltonian mechanics. In this paper the symplectic geometry structure of Birkhoffian system is discussed, then the symplecticity of Birkhoffian phase flow is presented. Based on these properties we give a way to construct symplectic schemes for Birkhoffian systems by using the generating function method. 展开更多
关键词 Birkhoffran system symplectic structure generating function method symplectic scheme
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POINCARE-CARTAN INTEGRAL INVARIANTS OF BIRKHOFFIAN SYSTEMS 被引量:3
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作者 郭永新 尚玫 罗绍凯 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2003年第1期68-72,共5页
Based on modern differential geometry, the symplectic structure of a Birkhoffian system which is an extension of conservative and nonconservative systems is analyzed. An integral invariant of Poincaré_Cartan... Based on modern differential geometry, the symplectic structure of a Birkhoffian system which is an extension of conservative and nonconservative systems is analyzed. An integral invariant of Poincaré_Cartan's type is constructed for Birkhoffian systems. Finally, one_dimensional damped vibration is taken as an illustrative example and an integral invariant of Poincaré's type is found. 展开更多
关键词 Birkhoffian systems symplectic structure self_adjointness Poincaré_Cartan integral invariants
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Research on the discrete variational method for a Birkhoffian system
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作者 刘世兴 花巍 郭永新 《Chinese Physics B》 SCIE EI CAS CSCD 2014年第6期328-332,共5页
In this paper, we present a new integration algorithm based on the discrete Pfaff-Birkhoff principle for Birkhoffian systems. It is proved that the new algorithm can preserve the general symplectic geometric structure... In this paper, we present a new integration algorithm based on the discrete Pfaff-Birkhoff principle for Birkhoffian systems. It is proved that the new algorithm can preserve the general symplectic geometric structures of Birkhoffian systems. A numerical experiment for a damping oscillator system is conducted. The result shows that the new algorithm can better simulate the energy dissipation than the R-K method, which illustrates that we can numerically solve the dynamical equations by the discrete variational method in a Birkhoffian framework for the systems with a general symplectic structure. Furthermore, it is demonstrated that the results of the numerical experiments are determined not by the constructing methods of Birkhoffian functions but by whether the numerical method can preserve the inherent nature of the dynamical system. 展开更多
关键词 Birkhoff's equations discrete variational methods general symplectic structure discrete Birkhoff's equations
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Symplectic feedback using Hamiltonian Lie algebra and its applications to an inverted pendulum
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作者 J. Lorca ESPIRO C. Munoz POBLETE 《控制理论与应用(英文版)》 EI CSCD 2013年第2期275-281,共7页
From the symplectic representation of an autonomous nonlinear dynamical system with holonomic con- straints, i.e., those that can be represented through a symplectic form derived from a Hamiltonian, we present a new p... From the symplectic representation of an autonomous nonlinear dynamical system with holonomic con- straints, i.e., those that can be represented through a symplectic form derived from a Hamiltonian, we present a new proof on the realization of the symplectic feedback action, which has several theoretical advantages in demonstrating the uniqueness and existence of this type of solution. Also, we propose a technique based on the interpretation, construction and character- ization of the pull-back differential on the symplectic manifold as a member of a one-parameter Lie group. This allows one to synthesize the control law that governs a certain system to achieve a desired behavior; and the method developed from this is applied to a classical system such as the inverted pendulum. 展开更多
关键词 Hamiltonian systems symplectic structures Lie algebra Nonlinear dynamical systems Geometric con-trol
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A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications
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作者 Zhiping Qiu Nan Jiang 《Advances in Applied Mathematics and Mechanics》 SCIE 2021年第6期1535-1557,共23页
In this paper,a novel symplectic conservative perturbation series expansion method is proposed to investigate the dynamic response of linear Hamiltonian systems accounting for perturbations,which mainly originate from... In this paper,a novel symplectic conservative perturbation series expansion method is proposed to investigate the dynamic response of linear Hamiltonian systems accounting for perturbations,which mainly originate from parameters dispersions and measurement errors.Taking the perturbations into account,the perturbed system is regarded as a modification of the nominal system.By combining the perturbation series expansion method with the deterministic linear Hamiltonian system,the solution to the perturbed system is expressed in the form of asymptotic series by introducing a small parameter and a series of Hamiltonian canonical equations to predict the dynamic response are derived.Eventually,the response of the perturbed system can be obtained successfully by solving these Hamiltonian canonical equations using the symplectic difference schemes.The symplectic conservation of the proposed method is demonstrated mathematically indicating that the proposed method can preserve the characteristic property of the system.The performance of the proposed method is evaluated by three examples compared with the Runge-Kutta algorithm.Numerical examples illustrate the superiority of the proposed method in accuracy and stability,especially symplectic conservation for solving linear Hamiltonian systems with perturbations and the applicability in structural dynamic response estimation. 展开更多
关键词 Linear Hamiltonian system perturbation series expansion method symplectic structure symplectic algorithm structural dynamic response
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