부산대학교 도서관

In this work, the authors provide an elementary introduction tothe notion of symplectic maps, their relationship to Hamiltoniansystems of ODEs, and their use as the basis for a symplecticintegration algorithm (SIA). Also, the fractals arising in the 3-bodyproblem are investigated. To do this, the simplified trilinear 3-bodyproblem is considered and an example of a symplectic integrationalgorithm is presented. The authors illustrate why an SIA is ideallysuited for generating the fractal structure which can arise in phasespace for Hamiltonian systems of equations. Taking into account thatmany approaches to designing SIAs involve the use of generatingfunctions or Lie algebraic techniques, the authors state that theirpresentation ofthis type of algorithm is more elementary and well suited forundergraduates.\parIn Section 2 the authors recall some notions about Hamiltonian mechanics,exemplified by the undamped, unforced harmonic oscillator andrespectively by the undamped, unforced ideal pendulum. Using thePoincaré-Bendixson Theorem, they conclude that at least 3 bodies areneeded to potentially have fractal geometry and chaotic dynamicspresent in the $n$-body problem.\parIn Section 3, the trilinear 3-body problem is presented. TheHamiltonian version of this problem is deduced.\parA presentation of symplectic maps is given in Section 4. The harmonicoscillator example is chosen to illustrate the fact that ``time-$t$''maps arising from Hamiltonian systems of ODEs are always symplectic.\parIn Section 5, the connection between Hamiltonian flows and symplecticmaps is used to conclude that symplectic maps have great potential applicationsto numerical integration of Hamiltonian systems of ODEs. Inparticular, symplectic maps are ideally suited for generating a global,qualitative structure in phase space for the trilinear 3-body problem.\parThe description of the SIA used to generate the images for thetrilinear 3-body model is given in Section 6. This particular algorithmconcerns separable Hamiltonians.\parIn the last section, some numerical results are given. A comparisonbetween the chosen symplectic integration algorithm and the fourth-orderRunge-Kutta algorithm is made. It provides striking visualevidence highlighting the superiority of the SIA to the Runge-Kuttaalgorithm. Also, in some pictures, fractal structure and chaoticdynamics in the trilinear 3-body problem are shown.