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Non-Linear Effects in Optical Systems by Lie Algebra and Symplectic Mapping

Journal of Modern Physics 2022年第11期13卷 1314-1323页

作者： Víctor M. Castañ o Centro de Física Aplicada y Tecnología Avanzada/Secretaría de Desarrollo Institucional Universidad Nacional Autónoma de México Queretaro México

The use of signals of different frequencies determines the geometrical deviation with respect to the optical axes of a given beam. This angle can be determined by Sympletic Map (SM), a powerful and simple mathematical tool for the characterization and construction of images in Geometrical Optics. The Sympletic Map constitutes a Lie Group, with an algebra associated: the Lie Algebra. In general, the SM can be expressed as an infinite series, where each term corresponds to different contributions produced by the optical devices that constitute the optical system (lenses, apertures, bandwidth cutoff, etc.). The level of correction to be performed on the image to recover the original object is clear and controllable by SM. This formalism can be extended easily to physical optics to describe diffraction and interference phenomena.

关键词： Symplectic Mapping Geometrical Optics Non-Linear Effects

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VARIATIONAL INTEGRATORS FOR HIGHER ORDER DIFFERENTIAL EQUATIONS

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Journal of Computational Mathematics 2003年第2期21卷 135-144页

作者： AajuanSun MengzheoQin InstituteofMathematics AcademyofMathematicsandSystemsSciencesChinsesAcademyofSciencesBeijing100080China StateKeyLaboratoryofScientific/EngineeringComputing InstituteofComputationalMathematicsandScientific/EngineeringComputingAcademyofMathematicsandSystemSciencesChineseAcademyofSciencesP.O.Box2719Beijing100080China

We analyze three one parameter families of approximations and show that they are symplectic in Lagrangian sence and can be related to symplectic schemes in Hamiltonian sense by different Symplectic Mappings. We also give a direct generalization of Veselov variational principle for construction of scheme of higher order differential equations. At last, we present numerical experiments.

关键词： Variational integrator Symplectic Mapping

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Quasiperiodic solutions of higher dimensional Duffing's equations via the KAM theorem

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Science China Mathematics 2001年第5期44卷 631-644页

作者：张福保 Department of Mathematics Nanjing University Nanjing China

In this paper, we consider the higher dimensional second order differential equations of the form x +　 F( x,t) = 0, x ∈Rn, with a class of weakly coupled potentials F(x,t), periodically depending on t. We prove the ... 详细信息

关键词： quasiperiodic solutions invariant tori Symplectic Mapping KAM theorem.

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DIFFUSION CHARACTERS OF THE ORBITS IN THE ASTEROID MOTION

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Applied Mathematics and Mechanics(English Edition) 2001年第7期22卷 808-819页

作者：周礼勇孙义燧周济林 Department of Astronomy Nanjing University Nanjing 210093 P R China

A Symplectic Mapping is studied carefully. The exponential diffusion last, in developed chaotic region and algebraic law in mixed region were observed. An area was found where the diffusion follows a logarithmic law. It is shown in the vicinity of an island, the logarithm of the escape time decreases linearily as the initial position moves away from the island. But when approaching close to the island, the escape time goes up very quickly, consistent with the superexponential stability of the invariant curve. When applied to the motion of asteroid, this mapping's fixed points and their stabilities give an explanation of the distribution of asteroids. The diffusion velocities in 4:3, 3:2 and 2:1 jovian resonances are also investigated.

关键词： diffusion Symplectic Mapping asteroid

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