Planar Four-body Solution, And Eight Issues Of A New Cycle

The N-body problem is a system of ordinary differential equations that describes the motion of N point masses or particles moving under Newton's law of motion, where the only acting forces are the mutual gravitational attractions .The problem is solved for N=2 because it can be reduced to the Kepler problem which is a system of ordinary differential equations that describes the motion of a particle moving under the gravitational attraction of a second particle fixed at the origin .The solutions of the Kepler problem are conic sections—circles ,ellipses , parabolas , and hyperbolas and straight line.Newton' s formulation of his laws of motion and his law of gravity was one of the greatest scientific accomplishments of all times. With these simple principles he was able to completely solve the two-body problem deriving Kepler'slaws describing the motion of the planet Mars . To the first approximation the orbit of Mars is solution of the two-body problem where only thegravitational forces of the sun and Mars are taken into account and this problem can be reduced to the Kepler problem.Newton next turned to the problem of describing the orbit of the moon. This is a harder problem since the first approximation should be a three-body problem—the earth, moon and sun.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. However, since Newton' s time there have been thousands of papers written on the N-body problem. These papers contain special solutions, asymptotic estimates, information about collision, the existence and non-existence of integrals, series solutions, non-collision singularities, etc.In this paper, we discuss the four-body and eight-body problem . In the first part, we prove the existence of the new non-collision periodic solution such that 4N(N = 1,2) bodies moving on 2N(N = l,2) orbits. Then we prove the existence of the new non-collision periodic solution such that 4N(N = 1,2) bodies moving on two orbits.