Planar Five-body And A New Cycle Of The Seven-body Problem Solution

The n-body problem is the problem of finding , given the initial positions , masses, and velocities of n bodies .their subsequent motions as determined by classical mechanics , i. e. Newton' s laws of motion and Newton' s law of gravity .Suppose that you have two bodies interacting only under the force of gravity (in Newtonian mechanics), and that you are given their positions, mass, and direction and speed of motion .You can then calculate rigorously their future movements. The solutions of the Kepler problem are conic sections—circles, ellipses, parabolas, and hyperbolas and straight line. However, if we add a third body, it turns out that there is no general method to calculate what will happen. Matters get more difficult if we allow many bodies (so the problem is sometimes called "the n-body problem"). No general solution to this problem exists. It is now widely believed that the N-body problem for N≥3 cannot be solved in the same sense as the two-body problem . In fact there is very good evidence that the general N-body problem is not solvable.In recent years variation methods play a more and more important role in n-body problem. Using variation methods a number of new orbits have been established over the last 5 years. Especially for three-body problem Alain Chenciner and Richard Montgomery found a new solution. Then using variation methods Prof Zhang S. Q. and Zhou Q. give a very short proof.In this paper, we discuss the five-body and seven-body problem by variation methods. In the first part, we prove the existence of the new non-collision periodic solution such that 3 bodies moving on one orbit with T = l and 2 bodies moving on another orbit with 7 = 1/3. Then we prove the existence of the new non-collision periodic solution such that 5 bodies moving on one orbit with T = 1 and 2 bodies moving on another orbit with...