## 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.

### MSC:

 70H05 Hamilton’s equations 70F07 Three-body problems 70F10 $$n$$-body problems 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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### References:

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