##
**Symmetries and noncollision closed orbits for planar N-body-type problems.**
*(English)*
Zbl 0715.70016

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

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### References:

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