Symmetries and noncollision closed orbits for planar N-body-type problems.

*(English)*Zbl 0715.70016In the last few years, calculus of variations in the large has been used to study Hamiltonian systems with singular potentials. In particular problems such as (0) \(m_ i\ddot y_ i+\nabla_{y_ i}V(y_ 0,...,y_ N)=0\), \(y_ i\in R^ N\), \(i=1,...,N\) have been studied, taking potentials V which have a suitable symmetry and, roughly, behave like \(V\approx \sum_{i<j}m_ im_ j/| y_ i-y_ j|^{\alpha}.\)

Here we deal with planar systems like (0). We prove, under suitable symmetry assumptions, the existence of a noncollision T-periodic solution (for any given \(T>0)\) of system (0) and we prove existence of a “new” solution of the three-body problem.

Section 2 is devoted to prove a class of estimates which hold under suitable symmetry assumptions (all verified by the N-body problem) and which will be the essential tool to rule out collisions.

In Section 3 we prove the existence of a noncollision solution for problem (0). We remark that this is the first result concerning existence of noncollision solutions for N-body-type problems without the “strong force” assumption.

Finally, in Section 4 we prove the existence of a “new” solution for the planar three-body problem. In particular the existence of a T- periodic solution is proven for the planar three-body problem (for a suitable choice of masses, which includes for example \(m_ 1=1\), \(m_ 2=1\), \(m_ 3=0.3\) and \(m_ 1=1\), \(m_ 2=300,000\), \(m_ 3=0.01)\) which is different from the Lagrange triangular solutions and the Euler collinear ones.

Here we deal with planar systems like (0). We prove, under suitable symmetry assumptions, the existence of a noncollision T-periodic solution (for any given \(T>0)\) of system (0) and we prove existence of a “new” solution of the three-body problem.

Section 2 is devoted to prove a class of estimates which hold under suitable symmetry assumptions (all verified by the N-body problem) and which will be the essential tool to rule out collisions.

In Section 3 we prove the existence of a noncollision solution for problem (0). We remark that this is the first result concerning existence of noncollision solutions for N-body-type problems without the “strong force” assumption.

Finally, in Section 4 we prove the existence of a “new” solution for the planar three-body problem. In particular the existence of a T- periodic solution is proven for the planar three-body problem (for a suitable choice of masses, which includes for example \(m_ 1=1\), \(m_ 2=1\), \(m_ 3=0.3\) and \(m_ 1=1\), \(m_ 2=300,000\), \(m_ 3=0.01)\) which is different from the Lagrange triangular solutions and the Euler collinear ones.

##### MSC:

70H05 | Hamilton’s equations |

70F07 | Three-body problems |

70F10 | \(n\)-body problems |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

##### Keywords:

Hamiltonian systems with singular potentials; noncollision solutions; “strong force” assumption; planar three-body problem
PDF
BibTeX
XML
Cite

\textit{U. Bessi} and \textit{V. Coti Zelati}, Nonlinear Anal., Theory Methods Appl. 16, No. 6, 587--598 (1991; Zbl 0715.70016)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Ambrosetti, A.; Coti Zelati, V., Critical points with lack of compactness and singular dynamical systems, Ann. mat. pura appl., 149, 237-259, (1987) · Zbl 0642.58017 |

[2] | Ambrosetti, A.; Coti Zelati, V., Perturbation of Hamiltonian systems with Keplerian potentials, Math. Z., 201, 201-242, (1989) · Zbl 0653.34032 |

[3] | Bahri, A.; Rabinowitz, P.H., A minimax method for a class of Hamiltonian systems with singular potentials, J. funct. analysis, 82, 412-428, (1989) · Zbl 0681.70018 |

[4] | Bahri, A.; Rabinowitz, P.H., Solutions of the three-body problem via critical points at infinity, (1988), University of Wisconsin-Madison, preprint |

[5] | C{\scOTI} Z{\scELATI} V., A class of periodic solutions of the N-body problem, Celestial Mech. (to appear). |

[6] | C{\scOTI} Z{\scELATI} V., Periodic solutions for N-body-type problems, Ann.Inst. H. Poincaré Analyse non Linéaire (to appear). |

[7] | C{\scOTI} Z{\scELATI} V., A class of periodic solutions for N-body-type dynamical systems, Appl. Math. Letters (to appear). |

[8] | D{\scEGIOVANNI} M. & G{\scIANNONI} F., Periodic solutions for dynamical systems with Newtonian-type potentials, Ann. Scu. norm. sup. Pisa Cl. Sci. (4) (to appear). |

[9] | Gordon, W., Conservative dynamical systems involving strong forces, Trans am. math. soc., 204, 113-135, (1975) · Zbl 0276.58005 |

[10] | Greco, C., Periodic solutions of a class of singular Hamiltonian systems, Nonlinear analysis, 12, 259-269, (1988) · Zbl 0648.34048 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.