The masses in a symmetric centered solution of the n-body problem. (English) Zbl 0693.70010
We saw recently [the author, Celestial Mech. Dyn. Astron. 46, No.1, 27-30 (1989; Zbl 0676.70011)] that if a noncollinear solution of the four body problem has a symmetry axis, then the center of mass of the particle system lies on this axis and the symmetric masses are equal. Following the proof of that paper it is easy to see that the property remains true for a solution of the n-body problem having two pairs of symmetric masses relative to an axis while the other n-4 particles belong, for all time, to this axis. Unfortunately, the method developed there cannot be used for more than two pairs of symmetric particles. We will give here a different way to treat this problem in the case of a planar solution with any number of symmetric bodies but imposing the restriction that the baricenter of the particle system lies (at least for an open interval of time) on the symmetry axis. Such a solution will be called symmetric centered. It is actually not known (excepting the particular case with only two symmetric pairs - discussed above) if any symmetric solution with respect to an axis is always symmetric centered, but this property might be true. Anyway, the author doesn’t know of any proof of this fact.
It will be shown that if a planar solution of the n-body problem leads to a motion that has at least one pair of symmetric particles relative to a symmetry axis, the other particles and the center of mass of the system belonging to the axis, then the symmetric particles have equal masses. In order to perform the proof we will also see that along a nonsymmetric solution, the set of time moments when symmetric configurations occur is formed by isolated points.
Finally we prove that the set of initial conditions leading to symmetric centered solutions is of measure of zero and nowhere dense relative to the set of all initial conditions that define solutions in \({\mathbb{R}}^ 2\). This means that such solutions are improbable in the sense that the set of initial conditions leading to them is poor from the measure theory and topological point of view.
It will be shown that if a planar solution of the n-body problem leads to a motion that has at least one pair of symmetric particles relative to a symmetry axis, the other particles and the center of mass of the system belonging to the axis, then the symmetric particles have equal masses. In order to perform the proof we will also see that along a nonsymmetric solution, the set of time moments when symmetric configurations occur is formed by isolated points.
Finally we prove that the set of initial conditions leading to symmetric centered solutions is of measure of zero and nowhere dense relative to the set of all initial conditions that define solutions in \({\mathbb{R}}^ 2\). This means that such solutions are improbable in the sense that the set of initial conditions leading to them is poor from the measure theory and topological point of view.
MSC:
| 70F10 | \(n\)-body problems |
| 70F15 | Celestial mechanics |
| 37-XX | Dynamical systems and ergodic theory |
| 28A75 | Length, area, volume, other geometric measure theory |
| 54F65 | Topological characterizations of particular spaces |
