Approximating the Invariant Sets of a Finite Straight Segment near Its Collinear Equilibria
Approximating the Invariant Sets of a Finite Straight Segment near Its Collinear Equilibria
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DOI：
10.1137/040614517
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发表时间：
2006-08
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影响因子：
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通讯作者：
J. Palacián;P. Yanguas;Susana Gutiérrez-Romero
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文献类型：
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作者：
J. Palacián;P. Yanguas;Susana Gutiérrez-Romero
本文讨论了一种明确计算m自由度哈密顿系统平衡点附近一些不变流形渐近表达式的方法。该方法基于广义范式的使用。我们的目标是通过将该技术应用于哈密顿系统的共线平衡点的研究来展示它的力量，哈密顿系统是由一个粒子绕有限直线段运动所定义的。我们使用了三种不同的李变换：(i)我们计算了共线性平衡中心流形对应的Hamilton函数；（ii）我们确定了与稳定-不稳定方向相关的哈密顿量；(3)得到与两个稳-稳方向之一相关的哈密顿量。通过(i)，我们可以利用两种变换的直接坐标变化来参数化原始系统的中心、稳定流形和不稳定流形。我们也……
This paper reviews a method for computing explicitly the asymptotic expressions of some invariant manifolds in the vicinity of equilibrium points corresponding to Hamiltonian systems of m degrees of freedom. The method is based on the use of generalized normal forms. Our goal is to show the power of this technique by applying it to the study of one of the colinear equilibrium points corresponding to the Hamiltonian system defined by the motion of a particle orbiting around a finite straight segment. We make use of three different Lie transformations: (i) we calculate the Hamilton function corresponding to the center manifold of the colinear equilibrium; (ii) we determine the Hamiltonian related to the stable-unstable direction; and (iii) we obtain the Hamiltonian related to one of the two stable-stable directions. By means of (i), we are able to parameterize the center, the stable, and the unstable manifolds of the original system using the direct changes of coordinates of the two transformations. We also...