Abstract
Under arbitrary masses, in this paper, we discuss the existence of new families of spatial central configurations for the N + N + 2‐body problem, N ≥ 2. We study some necessary conditions and sufficient conditions for a families of spatial double pyramidical central configurations (d.p.c.c.), where 2N bodies are at the vertices of a nested regular N‐gons Γ1 ∪ Γ2, and the other two bodies are symmetrically located on the straight line that is perpendicular to the plane that contains Γ1 ∪ Γ2 and passes through the geometric center of Γ1 ∪ Γ2. We prove that if the bodies are in a d.p.c.c., then the masses on each N‐gon are equal, and the other two are also equal. And also we prove the existence and uniqueness of the central configurations for any given ratios of masses.
Highlights
Main ResultsThe Newtonian n-body problem see 1–7 concerns the motion of n point particles with masses mj ∈ R and positions qj ∈ R3 j 1, . . . , n
We study new classes of spatial double pyramidical central configurations d.p.c.c for the N N 2-body that satisfy the following
The main results of this paper are the following
Summary
The Newtonian n-body problem see 1–7 concerns the motion of n point particles with masses mj ∈ R and positions qj ∈ R3 j 1, . . . , n. The Newtonian n-body problem see 1–7 concerns the motion of n point particles with masses mj ∈ R and positions qj ∈ R3 j 1, . The motion is governed by Newton’s law: mj qj. Qn and U q is the Newtonian potential: Uq mkmj ∂U q ∂qj , 1.1 where q q1, . . . , qn and U q is the Newtonian potential: Uq mkmj
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