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It is well known that the many-body problem is very important for a wide variety of applications, ranging from theoretical physics to celestial mechanics and astrodynamics. But the differential equations of this problem are in general not integrable. So, according to Poincare’s ideas, the further progress in this field will be connected with finding new classes of the exact particular solutions of the many-body problem and investigating their stability. But the stability investigation turned out to be the most complicated problem of qualitative theory of differential equations. For example, solving of the stability problem of the Lagrange triangular solutions has taken about 200 years, whereas stability of the homographic and homothetic solutions in the three-body problem still remains unsolved. However, now we have new computation systems, for example Mathematica that essentially increase our ability to do both numerical and symbolic calculations. So there is a hope that we can push considerably both the many-body problem and the theory of dynamical systems generally using modern computer algebra systems.
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