Apollonius coordinates, the N-body problem, and continuation of periodic solutions
The planar N-body problem is considered. With the help of any solution of the Kepler problem a coordinate system can be defined in which the Hamiltonian of the planar N-body problem is relatively simple. Author calls these symplectic coordinates as Apollonius coordinates. If the solution of the Kepler problem is elliptic, this coordinate system is the rotating-pulsating one used in the elliptic restricted three-body problem. To get Appolonius coordinates two symplectic coordinate changes (detailed in the paper) are necessary. The critical point of the Hamiltonian of the N-body problem in Apollonius coordinates is defined as a ``relative equilibrium''. Paper proves: relative equilibria are central configurations. Then the method of symplectic scaling is used to deal with the restricted \((N+1)\)-body problem and its connection with the full \((N+1)\)-body problem. The same method is applied to show: a nondegenerate periodic solution of the elliptic Hill's lunar equation can be continued into the full three-body problem. Due a symmetry of the N-body problem the algebraic multiplicity of the characteristic multiplier \(+1\) of a periodic solution of the N-body problem is at least 8. Paper shows possibilities for the reduction of this degeneracy. Then a continuation theorem is given in connection with the nondegenerate periodic solution of the elliptic, restricted \((N+1)\)-body problem into the full \((N+1)\)- body problem. The \((N+1)\)-body problem's Hamiltonian is presented also in Jacobi-Apollonius coordinates. Appolonius coordinates are applied to the two-body problem too.
scientific article; zbMATH DE number 4209657