Central Configurations, Periodic Orbits, and Hamiltonian Systems

New families of periodic orbits for a galactic potential

“…On the other hand, the fact that we cannot find periodic orbits when a ∈ {0, 2} using the averaging theory indicates that for these values of the parameter a the Hamiltonian system can be completely integrable, and consequently their periodic solutions are not isolated in the set of all periodic solutions by the Liouville-Arnold Theorem, and this is the case as it is proved in Proposition 4. For more details on the results stated in these paragraph see the books [1] and [8], the first for the results on Hamiltonian systems and the second for the results on averaging theory.…”

Generic Finiteness for a Class of Symmetric Planar Central Configurations of the Six-Body Problem and the Six-Vortex Problem

“…In this work, we only consider central configurations with six bodies and dimension two. An excellent introductory text about central configurations theory is [17,Ch.II].…”

On a Class of Central Configurations in the Planar $${\varvec{3n}}$$ 3 n -Body Problem

“…We add here also the case (1, 3) as an equilateral triangle with mass m 0 = 0 located at its center of gravity ( [3,10,12]). An interesting configuration of the type ( p, N ) can be (2,8), where eight point-masses are situated at the vertices of an equilateral triangle and a pentagon, which are concentric, (see, for example [13]). In our paper we deal with a special class of central configurations of the type (2, 3n), consisting of 3n bodies located at the vertices of two regular concentric polygons: the "interior" one of the radius q and with n equal point-masses, and other 2n bodies located at the vertices of the second regular 2n-gon of the radius, say, q , with point-masses of two categories: one half, i.e., n masses, all equal to, say m 2 , and another half, i.e.…”