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.


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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