The hierarchical approach to proving the existence of generalized planar nested central configurations on some versions of the general (pn+1)-body problem

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A hierarchical approach to proving of existence in the general (pn+1)-body exact partial solutions is presented, the so called generalized planar nested central configurations in a form of consequently nested in each other convex n -gons with nonequal in general masses in the vertices and a nonspherical body in the centre. Flat nest-shaped central configurations in the form of convex quadrilaterals of mixed shapes nested one into another of the type square + rhombus + deltoid + trapezoid + central body within the frame-work of the general problem of (4n+1)-bodies of celestial mechanics were measured. The given general conditions of existence are valid for any nest-shaped planar central configurations within the framework of the (4n+1)-bodies problem. Symbolic calculations of the Maple mathematical package are used to solve the system of equations. The system of algebraic equations has a hierarchical structure similar to the obtained direct transformations to the system of algebraic equations within the process of solving systems of linear equations by the Gauss method. The cases of a central body in the form of a spherical (a ball) and a non-spherical (an ellipsoid of rotation or a triaxial ellipsoid) structures are considered. In each of the cases, the corresponding necessary and sufficient conditions for the existence of central configurations of various types are given.

About the authors

Yulianna V. Perepelkina

Russian State University of Tourism and Service

Author for correspondence.
ORCID iD: 0000-0001-8115-8253
SPIN-code: 5157-4093
Scopus Author ID: 25925321600

Candidate of Physical and Mathematical Sciences, Assistant Professor of High School of Service

99 Glavnaya St, Cherkizovo, 141221, Russian Federation

Alexander N. Zadiranov

State Fire Academy of EMERCOM of Russia

ORCID iD: 0000-0001-7787-8290
SPIN-code: 2873-6465
Scopus Author ID: 57214856655

Doctor of Technical Sciences, Professor of Combustion Behavior and Environmental Safety Department, Educational and Scientific Complex of Combustion Processes and Environmental Safety

4 Borisa Galushkina St, Moscow, 129366, Russian Federation


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Copyright (c) 2023 Perepelkina Y.V., Zadiranov A.N.

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