第二保群結構法是一種新的求解非線性動態系統的數值方法。先將要計算的非線性動態 系統轉為增廣動態系統,並且在迭代過程中,同時維持住群的特性。此數值方法透過簡單的 符號函數來描述動態系統的複雜性。在圓周運動中,第二保群結構法具有較優異的表現。因 此,我們利用此數值方法處理四星體問題中的三種圓周運動,並與其他數值方法做比較,藉 此測試其優異程度。此外,科學家們近期發現一種新的週期性運動:4-chains結構,我們也將 其加入討論。

The second group-preserving scheme GPS2 is a new numerical method to deal with nonlinear dy- namical system. It casts the nonlinear dynamical system into an augmented dynamical system, and the group properties is preserved for every iteration at the same time. This numerical method describes the complexity of the dynamical system by a simple signum function. The second group-preserving scheme has a good performance in circular motions, so we compute three circular motions, and com- pare with other numerical methods to test this scheme. Moreover, we will compute a new periodic motions, 4-chains configuration, which is founded recently by scientists, and discuss about it.

論文審定書 i
摘要 ii
Abstract iii
1 Numerical method 1
1.1 Second group-preserving scheme 1
1.2 Numerical algorithm 9
2 Four-body problem 11
2.1 Dynamic system of celestial mechanic 11
2.2 Four-body orbit 11
3 Preliminary 13
3.1 Collinear configuration 13
3.2 Regular triangular configuration 16
3.3 Square configuration 18
3.4 4-chains configuration 20
4 Numerical results 21
4.1 Collinear configuration 21
4.2 Regular triangular configuration 24
4.3 Square configuration 27
4.4 4-chains configuration 30
5 Conclusion 33
Reference 34

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