부산대학교 도서관

In this work, the authors provide an elementary introduction to the notion of symplectic maps, their relationship to Hamiltonian systems of ODEs, and their use as the basis for a symplectic integration algorithm (SIA). Also, the fractals arising in the 3-body problem are investigated. To do this, the simplified trilinear 3-body problem is considered and an example of a symplectic integration algorithm is presented. The authors illustrate why an SIA is ideally suited for generating the fractal structure which can arise in phase space for Hamiltonian systems of equations. Taking into account that many approaches to designing SIAs involve the use of generating functions or Lie algebraic techniques, the authors state that their presentation of this type of algorithm is more elementary and well suited for undergraduates. \par In Section 2 the authors recall some notions about Hamiltonian mechanics, exemplified by the undamped, unforced harmonic oscillator and respectively by the undamped, unforced ideal pendulum. Using the Poincaré-Bendixson Theorem, they conclude that at least 3 bodies are needed to potentially have fractal geometry and chaotic dynamics present in the $n$-body problem. \par In Section 3, the trilinear 3-body problem is presented. The Hamiltonian version of this problem is deduced. \par A presentation of symplectic maps is given in Section 4. The harmonic oscillator example is chosen to illustrate the fact that ``time-$t$'' maps arising from Hamiltonian systems of ODEs are always symplectic. \par In Section 5, the connection between Hamiltonian flows and symplectic maps is used to conclude that symplectic maps have great potential applications to numerical integration of Hamiltonian systems of ODEs. In particular, symplectic maps are ideally suited for generating a global, qualitative structure in phase space for the trilinear 3-body problem. \par The description of the SIA used to generate the images for the trilinear 3-body model is given in Section 6. This particular algorithm concerns separable Hamiltonians. \par In the last section, some numerical results are given. A comparison between the chosen symplectic integration algorithm and the fourth-order Runge-Kutta algorithm is made. It provides striking visual evidence highlighting the superiority of the SIA to the Runge-Kutta algorithm. Also, in some pictures, fractal structure and chaotic dynamics in the trilinear 3-body problem are shown.