中国物理B ›› 2005, Vol. 14 ›› Issue (4): 660-662.doi: 10.1088/1009-1963/14/4/004

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The parametric orbits and the form invariance of three-body in one-dimension

楼智美   

  1. Department of Physics, Shaoxing College of Arts and Sciences,Shaoxing,312000, China
  • 收稿日期:2004-10-10 修回日期:2004-11-03 出版日期:2005-04-20 发布日期:2005-03-28

The parametric orbits and the form invariance of three-body in one-dimension

Lou Zhi-Mei (楼智美)   

  1. Department of Physics, Shaoxing College of Arts and Sciences, Shaoxing 312000, China
  • Received:2004-10-10 Revised:2004-11-03 Online:2005-04-20 Published:2005-03-28

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摘要： In this paper, the differential equations of motion of a three-body interacting pairwise by inverse cubic forces (``centrifugal potential'') in addition to linear forces (``harmonical potential'') are expressed in Ermakov formalism in two-dimension polar coordinates, and the Ermakov invariant is obtained. By rescaling of the time variable and the space coordinates, the parametric orbits of the three bodies are expressed in terms of relative energy H11 and Ermakov invariant. The form invariance of the transformations of two conserved quantities are also studied.

Abstract: In this paper, the differential equations of motion of a three-body interacting pairwise by inverse cubic forces (``centrifugal potential'') in addition to linear forces (``harmonical potential'') are expressed in Ermakov formalism in two-dimension polar coordinates, and the Ermakov invariant is obtained. By rescaling of the time variable and the space coordinates, the parametric orbits of the three bodies are expressed in terms of relative energy H1 and Ermakov invariant. The form invariance of the transformations of two conserved quantities are also studied.

Key words: three-body, parametric orbits, Ermakov invariant, Hamiltonian function, conserved quan-tities, form invariance

中图分类号:  (Few- and many-body systems ?)

  • 45.50.Jf
45.05.+x (General theory of classical mechanics of discrete systems) 45.20.Jj (Lagrangian and Hamiltonian mechanics)