Periodic trajectories for a two-dimensional nonintegrable Hamiltonian

• Published in: Annals of physics 1987-08, Vol.177 (2), p.330-358
• Main Authors: Baranger, M, Davies, K.T.R
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Geometry, differential geometry, and topology; Mathematical methods in physics; Physics
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/0003-4916(87)90123-0

Anumerical study is made of the classical periodic trajectories for the two-dimensional, nonintegrable Hamiltonian H= 1 2 (p 2 x+p 2 y)+(y− 1 2 x 2) 2+0.05x 2 . In addition to x− y pictures of the trajectories, E− τ (energy-period) plots of the periodie families are presented. Efforts have been made to include all trajectories with short periods and all simple branchings of these trajectories. The monodromy matrix has been calculated in all cases, and from it the stability properties are derived. The topology of the E− τ plot has been explored, with the following results. One family may have several stable regions. The plot is not completely connected; there are islands. The plot is not a tree; there are cycles. There are isochronous branchings, period-doublings, and period-multiplyings of higher orders, and examples of each of these are presented. There is often more than one branch issuing from a branch point. Some general empirical rules are inferred. In particular, the exitence of isochronous branchings is seen to be a consequence of the symmetry of the Hamiltonian. All these results agree with the general classification of possible branchings derived in Ref. [10]. ( M. A. M. de Aguiar, C. P. Malta, M. Baranger, and K. T. R. Davies, in preparation). Finally, some nonperiodic trajectories are calculated to illustrate the fact that stable periodic trajectories lie in “regular” regions of phase space, while unstable ones lie in “chaotic” regions.